10.4 Trigonometric Identities
|
|
- Ὕδρα Κασιδιάρης
- 7 χρόνια πριν
- Προβολές:
Transcript
1 770 Foundations of Trigonometry 0. Trigonometric Identities In Section 0.3, we saw the utility of the Pythagorean Identities in Theorem 0.8 along with the Quotient and Reciprocal Identities in Theorem 0.6. Not only did these identities help us compute the values of the circular functions for angles, they were also useful in simplifying expressions involving the circular functions. In this section, we introduce several collections of identities which have uses in this course and beyond. Our first set of identities is the Even / Odd identities. Theorem 0.. Even / Odd Identities: For all applicable angles, cos( cos( sec( sec( sin( sin( csc( csc( tan( tan( cot( cot( In light of the Quotient and Reciprocal Identities, Theorem 0.6, it suffices to show cos( cos( and sin( sin(. The remaining four circular functions can be expressed in terms of cos( and sin( so the proofs of their Even / Odd Identities are left as exercises. Consider an angle plotted in standard position. Let 0 be the angle coterminal with with 0 0 < π. (We can construct the angle 0 by rotating counter-clockwise from the positive x-axis to the terminal side of as pictured below. Since and 0 are coterminal, cos( cos( 0 and sin( sin( 0. y y 0 0 P (cos( 0, sin( 0 x Q(cos( 0, sin( 0 x 0 We now consider the angles and 0. Since is coterminal with 0, there is some integer k so that 0 + π k. Therefore, 0 π k 0 + π ( k. Since k is an integer, so is ( k, which means is coterminal with 0. Hence, cos( cos( 0 and sin( sin( 0. Let P and Q denote the points on the terminal sides of 0 and 0, respectively, which lie on the Unit Circle. By definition, the coordinates of P are (cos( 0, sin( 0 and the coordinates of Q are (cos( 0, sin( 0. Since 0 and 0 sweep out congruent central sectors of the Unit Circle, it As mentioned at the end of Section 0., properties of the circular functions when thought of as functions of angles in radian measure hold equally well if we view these functions as functions of real numbers. Not surprisingly, the Even / Odd properties of the circular functions are so named because they identify cosine and secant as even functions, while the remaining four circular functions are odd. (See Section.6.
2 0. Trigonometric Identities 77 follows that the points P and Q are symmetric about the x-axis. Thus, cos( 0 cos( 0 and sin( 0 sin( 0. Since the cosines and sines of 0 and 0 are the same as those for and, respectively, we get cos( cos( and sin( sin(, as required. The Even / Odd Identities are readily demonstrated using any of the common angles noted in Section 0.. Their true utility, however, lies not in computation, but in simplifying expressions involving the circular functions. In fact, our next batch of identities makes heavy use of the Even / Odd Identities. Theorem 0.3. Sum and Difference Identities for Cosine: For all angles α and β, cos(α + β cos(α cos(β sin(α sin(β cos(α β cos(α cos(β + sin(α sin(β We first prove the result for differences. As in the proof of the Even / Odd Identities, we can reduce the proof for general angles α and β to angles α 0 and β 0, coterminal with α and β, respectively, each of which measure between 0 and π radians. Since α and α 0 are coterminal, as are β and β 0, it follows that α β is coterminal with α 0 β 0. Consider the case below where α 0 β 0. y y P (cos(α 0, sin(α 0 α 0 β 0 Q(cos(β 0, sin(β 0 A(cos(α 0 β 0, sin(α 0 β 0 α 0 β 0 α 0 β 0 O x O B(, 0 x Since the angles P OQ and AOB are congruent, the distance between P and Q is equal to the distance between A and B. The distance formula, Equation., yields (cos(α0 cos(β 0 + (sin(α 0 sin(β 0 (cos(α 0 β 0 + (sin(α 0 β 0 0 Squaring both sides, we expand the left hand side of this equation as (cos(α 0 cos(β 0 + (sin(α 0 sin(β 0 cos (α 0 cos(α 0 cos(β 0 + cos (β 0 + sin (α 0 sin(α 0 sin(β 0 + sin (β 0 cos (α 0 + sin (α 0 + cos (β 0 + sin (β 0 cos(α 0 cos(β 0 sin(α 0 sin(β 0 In the picture we ve drawn, the triangles P OQ and AOB are congruent, which is even better. However, α 0 β 0 could be 0 or it could be π, neither of which makes a triangle. It could also be larger than π, which makes a triangle, just not the one we ve drawn. You should think about those three cases.
3 77 Foundations of Trigonometry From the Pythagorean Identities, cos (α 0 + sin (α 0 and cos (β 0 + sin (β 0, so (cos(α 0 cos(β 0 + (sin(α 0 sin(β 0 cos(α 0 cos(β 0 sin(α 0 sin(β 0 Turning our attention to the right hand side of our equation, we find (cos(α 0 β 0 + (sin(α 0 β 0 0 cos (α 0 β 0 cos(α 0 β sin (α 0 β 0 + cos (α 0 β 0 + sin (α 0 β 0 cos(α 0 β 0 Once again, we simplify cos (α 0 β 0 + sin (α 0 β 0, so that (cos(α 0 β 0 + (sin(α 0 β 0 0 cos(α 0 β 0 Putting it all together, we get cos(α 0 cos(β 0 sin(α 0 sin(β 0 cos(α 0 β 0, which simplifies to: cos(α 0 β 0 cos(α 0 cos(β 0 + sin(α 0 sin(β 0. Since α and α 0, β and β 0 and α β and α 0 β 0 are all coterminal pairs of angles, we have cos(α β cos(α cos(β + sin(α sin(β. For the case where α 0 β 0, we can apply the above argument to the angle β 0 α 0 to obtain the identity cos(β 0 α 0 cos(β 0 cos(α 0 + sin(β 0 sin(α 0. Applying the Even Identity of cosine, we get cos(β 0 α 0 cos( (α 0 β 0 cos(α 0 β 0, and we get the identity in this case, too. To get the sum identity for cosine, we use the difference formula along with the Even/Odd Identities cos(α + β cos(α ( β cos(α cos( β + sin(α sin( β cos(α cos(β sin(α sin(β We put these newfound identities to good use in the following example. Example Find the exact value of cos (.. Verify the identity: cos sin(. Solution.. In order to use Theorem 0.3 to find cos (, we need to write as a sum or difference of angles whose cosines and sines we know. One way to do so is to write 30. cos ( cos ( 30 cos ( cos (30 + sin ( sin (30 ( ( ( (
4 0. Trigonometric Identities 773. In a straightforward application of Theorem 0.3, we find cos cos cos ( + sin sin ( (0 (cos( + ( (sin( sin( The identity verified in Example 0.., namely, cos sin(, is the first of the celebrated cofunction identities. These identities were first hinted at in Exercise 7 in Section 0.. From sin( cos, we get: [ π ] sin cos cos(, which says, in words, that the co sine of an angle is the sine of its co mplement. Now that these identities have been established for cosine and sine, the remaining circular functions follow suit. The remaining proofs are left as exercises. Theorem 0.. Cofunction Identities: For all applicable angles, cos sin( sec csc( tan cot( sin cos( csc sec( cot tan( With the Cofunction Identities in place, we are now in the position to derive the sum and difference formulas for sine. To derive the sum formula for sine, we convert to cosines using a cofunction identity, then expand using the difference formula for cosine sin(α + β cos (α + β ([ π ] cos α β cos α cos(β + sin α sin(β sin(α cos(β + cos(α sin(β We can derive the difference formula for sine by rewriting sin(α β as sin(α + ( β and using the sum formula and the Even / Odd Identities. Again, we leave the details to the reader. Theorem 0.. Sum and Difference Identities for Sine: For all angles α and β, sin(α + β sin(α cos(β + cos(α sin(β sin(α β sin(α cos(β cos(α sin(β
5 77 Foundations of Trigonometry Example Find the exact value of sin ( 9π. If α is a Quadrant II angle with sin(α 3, and β is a Quadrant III angle with tan(β, find sin(α β. 3. Derive a formula for tan(α + β in terms of tan(α and tan(β. Solution.. As in Example 0.., we need to write the angle 9π as a sum or difference of common angles. The denominator of suggests a combination of angles with denominators 3 and. One such combination is 9π π 3 + π. Applying Theorem 0., we get ( ( 9π π sin sin 3 + π ( π ( π sin cos + cos sin 3 3 ( ( ( 3 + ( 6. In order to find sin(α β using Theorem 0., we need to find cos(α and both cos(β and sin(β. To find cos(α, we use the Pythagorean Identity cos (α + sin (α. Since sin(α 3, we have cos (α + ( 3, or cos(α ± 3. Since α is a Quadrant II angle, cos(α 3. We now set about finding cos(β and sin(β. We have several ways to proceed, but the Pythagorean Identity + tan (β sec (β is a quick way to get sec(β, and hence, cos(β. With tan(β, we get + sec (β so that sec(β ±. Since β is a Quadrant III angle, we choose sec(β so cos(β sec(β. We now need to determine sin(β. We could use The Pythagorean Identity cos (β + sin (β, but we opt instead to use a quotient identity. From tan(β sin(β cos(β, we have sin(β tan(β cos(β ( so we get sin(β (. We now have all the pieces needed to find sin(α β: sin(α β sin(α cos(β cos(α sin(β ( ( ( (
6 0. Trigonometric Identities We can start expanding tan(α + β using a quotient identity and our sum formulas tan(α + β sin(α + β cos(α + β sin(α cos(β + cos(α sin(β cos(α cos(β sin(α sin(β Since tan(α sin(α sin(β cos(α and tan(β cos(β, it looks as though if we divide both numerator and denominator by cos(α cos(β we will have what we want tan(α + β sin(α cos(β + cos(α sin(β cos(α cos(β sin(α sin(β cos(α cos(β cos(α cos(β sin(α cos(β cos(α sin(β + cos(α cos(β cos(α cos(β cos(α cos(β sin(α sin(β cos(α cos(β cos(α cos(β sin(α cos(β cos(α cos(β + cos(α sin(β cos(α cos(β cos(α cos(β sin(α sin(β cos(α cos(β cos(α cos(β tan(α + tan(β tan(α tan(β Naturally, this formula is limited to those cases where all of the tangents are defined. The formula developed in Exercise 0.. for tan(α+β can be used to find a formula for tan(α β by rewriting the difference as a sum, tan(α+( β, and the reader is encouraged to fill in the details. Below we summarize all of the sum and difference formulas for cosine, sine and tangent. Theorem 0.6. Sum and Difference Identities: For all applicable angles α and β, cos(α ± β cos(α cos(β sin(α sin(β sin(α ± β sin(α cos(β ± cos(α sin(β tan(α ± β tan(α ± tan(β tan(α tan(β In the statement of Theorem 0.6, we have combined the cases for the sum + and difference of angles into one formula. The convention here is that if you want the formula for the sum + of
7 776 Foundations of Trigonometry two angles, you use the top sign in the formula; for the difference,, use the bottom sign. For example, tan(α tan(β tan(α β + tan(α tan(β If we specialize the sum formulas in Theorem 0.6 to the case when α β, we obtain the following Double Angle Identities. Theorem 0.7. Double Angle Identities: For all applicable angles, cos ( sin ( cos( cos ( sin ( sin( sin( cos( tan( tan( tan ( The three different forms for cos( can be explained by our ability to exchange squares of cosine and sine via the Pythagorean Identity cos ( + sin ( and we leave the details to the reader. It is interesting to note that to determine the value of cos(, only one piece of information is required: either cos( or sin(. To determine sin(, however, it appears that we must know both sin( and cos(. In the next example, we show how we can find sin( knowing just one piece of information, namely tan(. Example Suppose P ( 3, lies on the terminal side of when is plotted in standard position. Find cos( and sin( and determine the quadrant in which the terminal side of the angle lies when it is plotted in standard position.. If sin( x for π π, find an expression for sin( in terms of x. 3. Verify the identity: sin( tan( + tan (.. Express cos(3 as a polynomial in terms of cos(. Solution.. Using Theorem 0.3 from Section 0. with x 3 and y, we find r x + y. Hence, cos( 3 and sin(. Applying Theorem 0.7, we get cos( cos ( sin ( ( 3 ( 7, and sin( sin( cos( ( ( 3. Since both cosine and sine of are negative, the terminal side of, when plotted in standard position, lies in Quadrant III.
8 0. Trigonometric Identities 777. If your first reaction to sin( x is No it s not, cos( x! then you have indeed learned something, and we take comfort in that. However, context is everything. Here, x is just a variable - it does not necessarily represent the x-coordinate of the point on The Unit Circle which lies on the terminal side of, assuming is drawn in standard position. Here, x represents the quantity sin(, and what we wish to know is how to express sin( in terms of x. We will see more of this kind of thing in Section 0.6, and, as usual, this is something we need for Calculus. Since sin( sin( cos(, we need to write cos( in terms of x to finish the problem. We substitute x sin( into the Pythagorean Identity, cos ( + sin (, to get cos ( + x, or cos( ± x. Since π π, cos( 0, and thus cos( x. Our final answer is sin( sin( cos( x x. 3. We start with the right hand side of the identity and note that + tan ( sec (. From this point, we use the Reciprocal and Quotient Identities to rewrite tan( and sec( in terms of cos( and sin(: ( sin( tan( + tan tan( ( sin( ( sec ( cos ( cos( cos( cos ( ( sin( cos( cos( cos( sin( cos( sin(. In Theorem 0.7, one of the formulas for cos(, namely cos( cos (, expresses cos( as a polynomial in terms of cos(. We are now asked to find such an identity for cos(3. Using the sum formula for cosine, we begin with cos(3 cos( + cos( cos( sin( sin( Our ultimate goal is to express the right hand side in terms of cos( only. We substitute cos( cos ( and sin( sin( cos( which yields cos(3 cos( cos( sin( sin( ( cos ( cos( ( sin( cos( sin( cos 3 ( cos( sin ( cos( Finally, we exchange sin ( for cos ( courtesy of the Pythagorean Identity, and get and we are done. cos(3 cos 3 ( cos( sin ( cos( cos 3 ( cos( ( cos ( cos( cos 3 ( cos( cos( + cos 3 ( cos 3 ( 3 cos(
9 778 Foundations of Trigonometry In the last problem in Example 0..3, we saw how we could rewrite cos(3 as sums of powers of cos(. In Calculus, we have occasion to do the reverse; that is, reduce the power of cosine and sine. Solving the identity cos( cos ( for cos ( and the identity cos( sin ( for sin ( results in the aptly-named Power Reduction formulas below. Theorem 0.8. Power Reduction Formulas: For all angles, cos ( + cos( sin ( cos( Example 0... Rewrite sin ( cos ( as a sum and difference of cosines to the first power. Solution. We begin with a straightforward application of Theorem 0.8 sin ( cos ( ( ( cos( + cos( ( cos ( cos ( Next, we apply the power reduction formula to cos ( to finish the reduction sin ( cos ( cos ( ( + cos(( 8 8 cos( 8 8 cos( Another application of the Power Reduction Formulas is the Half Angle Formulas. To start, we apply the Power Reduction Formula to cos ( cos ( + cos ( ( + cos(. We can obtain a formula for cos ( by extracting square roots. In a similar fashion, we may obtain a half angle formula for sine, and by using a quotient formula, obtain a half angle formula for tangent. We summarize these formulas below.
10 0. Trigonometric Identities 779 Theorem 0.9. Half Angle Formulas: For all applicable angles, ( + cos( cos ± ( cos( sin ± ( cos( tan ± + cos( where the choice of ± depends on the quadrant in which the terminal side of lies. Example Use a half angle formula to find the exact value of cos (.. Suppose π 0 with cos( 3. Find sin (. 3. Use the identity given in number 3 of Example 0..3 to derive the identity ( tan sin( + cos( Solution.. To use the half angle formula, we note that 30 and since is a Quadrant I angle, its cosine is positive. Thus we have + cos (30 cos ( Back in Example 0.., we found cos ( by using the difference formula for cosine. In that case, we determined cos ( 6+. The reader is encouraged to prove that these two expressions are equal.. If π 0, then π 0, which means sin ( < 0. Theorem 0.9 gives ( ( cos ( 3 sin
11 780 Foundations of Trigonometry 3. Instead of our usual approach to verifying identities, namely starting with one side of the equation and trying to transform it into the other, we will start with the identity we proved in number 3 of Example 0..3 and manipulate it into the identity we are asked to prove. The identity we are asked to start with is sin( tan(. If we are to use this to derive an identity for tan ( +tan (, it seems reasonable to proceed by replacing each occurrence of with sin ( ( tan ( + tan ( tan ( sin( + tan ( We now have the sin( we need, but we somehow need to get a factor of + cos( involved. To get cosines involved, recall that + tan ( sec (. We continue to manipulate our given identity by converting secants to cosines and using a power reduction formula tan ( sin( + tan ( sin( tan ( sec ( sin( tan ( cos ( sin( tan ( ( + cos ( ( sin( tan ( ( + cos( ( sin( tan + cos( Our next batch of identities, the Product to Sum Formulas, 3 are easily verified by expanding each of the right hand sides in accordance with Theorem 0.6 and as you should expect by now we leave the details as exercises. They are of particular use in Calculus, and we list them here for reference. Theorem 0.0. Product to Sum Formulas: For all angles α and β, cos(α cos(β [cos(α β + cos(α + β] sin(α sin(β [cos(α β cos(α + β] sin(α cos(β [sin(α β + sin(α + β] 3 These are also known as the Prosthaphaeresis Formulas and have a rich history. The authors recommend that you conduct some research on them as your schedule allows.
12 0. Trigonometric Identities 78 Related to the Product to Sum Formulas are the Sum to Product Formulas, which we will have need of in Section 0.7. These are easily verified using the Product to Sum Formulas, and as such, their proofs are left as exercises. Theorem 0.. Sum to Product Formulas: For all angles α and β, ( ( α + β α β cos(α + cos(β cos cos ( ( α + β α β cos(α cos(β sin sin ( ( α ± β α β sin(α ± sin(β sin cos Example Write cos( cos(6 as a sum.. Write sin( sin(3 as a product. Solution.. Identifying α and β 6, we find cos( cos(6 [cos( 6 + cos( + 6] cos( + cos(8 cos( + cos(8, where the last equality is courtesy of the even identity for cosine, cos( cos(.. Identifying α and β 3 yields ( 3 sin( sin(3 sin cos sin ( cos ( sin ( cos (, ( + 3 where the last equality is courtesy of the odd identity for sine, sin( sin(. The reader is reminded that all of the identities presented in this section which regard the circular functions as functions of angles (in radian measure apply equally well to the circular (trigonometric functions regarded as functions of real numbers. In Exercises 38-3 in Section 0., we see how some of these identities manifest themselves geometrically as we study the graphs of the these functions. In the upcoming Exercises, however, you need to do all of your work analytically without graphs.
13 78 Foundations of Trigonometry 0.. Exercises In Exercises - 6, use the Even / Odd Identities to verify the identity. Assume all quantities are defined.. sin(3π sin( 3π. cos ( π ( t cos t + π 3. tan( t + tan(t. csc( csc( +. sec( 6t sec(6t 6. cot(9 7 cot(7 9 In Exercises 7 -, use the Sum and Difference Identities to find the exact value. You may have need of the Quotient, Reciprocal or Even / Odd Identities as well. 7. cos(7 8. sec(6 9. sin(0 0. csc(9. cot(. tan(37 ( ( 3π π 3. cos. sin ( ( 7π 7π 6. cos 7. tan ( ( π π 9. cot 0. csc. If α is a Quadrant IV angle with cos(α, and sin(β. tan ( 3π 8. sin (. sec π 0 0, where π (a cos(α + β (b sin(α + β (c tan(α + β (d cos(α β (e sin(α β (f tan(α β < β < π, find 3. If csc(α 3, where 0 < α < π, and β is a Quadrant II angle with tan(β 7, find (a cos(α + β (b sin(α + β (c tan(α + β (d cos(α β (e sin(α β (f tan(α β. If sin(α 3, where 0 < α < π, and cos(β 3 where 3π < β < π, find (a sin(α + β (b cos(α β (c tan(α β
14 0. Trigonometric Identities 783. If sec(α 3, where π < α < π, and tan(β 7, where π < β < 3π, find (a csc(α β (b sec(α + β (c cot(α + β In Exercises 6-38, verify the identity. 6. cos( π cos( 7. sin(π sin( ( 8. tan + π cot( 9. sin(α + β + sin(α β sin(α cos(β 30. sin(α + β sin(α β cos(α sin(β 3. cos(α + β + cos(α β cos(α cos(β 3. cos(α + β cos(α β sin(α sin(β 33. sin(α + β + cot(α tan(β sin(α β cot(α tan(β cos(α + β tan(α tan(β cos(α β + tan(α tan(β sin(t + h sin(t h cos(t + h cos(t h tan(t + h tan(t h tan(α + β 3. tan(α β ( ( sin(h cos(h cos(t + sin(t h h ( cos(h cos(t h sin(t ( ( tan(h sec (t h tan(t tan(h ( sin(h h sin(α cos(α + sin(β cos(β sin(α cos(α sin(β cos(β In Exercises 39-8, use the Half Angle Formulas to find the exact value. You may have need of the Quotient, Reciprocal or Even / Odd Identities as well. 39. cos(7 (compare with Exercise 7 0. sin(0 (compare with Exercise 9. cos(67.. sin(7. ( 7π 3. tan(.. cos (compare with Exercise 6. sin (compare with Exercise 8 6. cos 8 7. sin ( π 8 8. tan ( 7π 8
15 78 Foundations of Trigonometry In Exercises 9-8, use the given information about to find the exact values of sin( ( sin 9. sin( 7 where 3π cos( ( cos tan( ( tan < < π 0. cos( 8 3 where 0 < < π. tan( where π < < 3π. csc( where π < < π 3. cos( 3 where 0 < < π. sin( where π < < 3π. cos( 3 where 3π < < π 6. sin( 3 where π < < π 7. sec( where 3π < < π 8. tan( where π < < π In Exercises 9-73, verify the identity. Assume all quantities are defined. 9. (cos( + sin( + sin( 60. (cos( sin( sin( 6. tan( tan( + tan( 6. csc( cot( + tan( sin ( cos( cos( cos ( cos( + cos( sin(3 3 sin( sin 3 ( 66. sin( sin( cos 3 ( sin 3 ( cos( sin ( cos ( + cos( cos( cos( sin ( cos ( cos( cos( + cos(6 69. cos( 8 cos ( 8 cos ( cos(8 8 cos 8 ( 6 cos 6 ( + 60 cos ( 3 cos ( + (HINT: Use the result to sec( cos( cos( + sin( + sin( cos( sin( cos( sin( + cos( + sin( cos( cos( cos( sin( cos( + sin( sin( cos(
16 0. Trigonometric Identities 78 In Exercises 7-79, write the given product as a sum. You may need to use an Even/Odd Identity. 7. cos(3 cos( 7. sin( sin(7 76. sin(9 cos( 77. cos( cos(6 78. sin(3 sin( 79. cos( sin(3 In Exercises 80-8, write the given sum as a product. You may need to use an Even/Odd or Cofunction Identity. 80. cos(3 + cos( 8. sin( sin(7 8. cos( cos(6 83. sin(9 sin( 8. sin( + cos( 8. cos( sin( 86. Suppose is a Quadrant I angle with sin( x. Verify the following formulas (a cos( x (b sin( x x (c cos( x 87. Discuss with your classmates how each of the formulas, if any, in Exercise 86 change if we change assume is a Quadrant II, III, or IV angle. 88. Suppose is a Quadrant I angle with tan( x. Verify the following formulas (a cos( x + (b sin( x x + (c sin( x x + (d cos( x x Discuss with your classmates how each of the formulas, if any, in Exercise 88 change if we change assume is a Quadrant II, III, or IV angle. 90. If sin( x for π < < π, find an expression for cos( in terms of x. 9. If tan( x 7 for π < < π, find an expression for sin( in terms of x. 9. If sec( x for 0 < < π, find an expression for ln sec( + tan( in terms of x. 93. Show that cos ( sin ( cos ( sin ( for all. 9. Let be a Quadrant III angle with cos(. Show that this is not enough information to ( determine the sign of sin by first assuming 3π < < 7π and then assuming π < < 3π ( and computing sin in both cases.
17 786 Foundations of Trigonometry 9. Without using your calculator, show that In part of Example 0..3, we wrote cos(3 as a polynomial in terms of cos(. In Exercise 69, we had you verify an identity which expresses cos( as a polynomial in terms of cos(. Can you find a polynomial in terms of cos( for cos(? cos(6? Can you find a pattern so that cos(n could be written as a polynomial in cosine for any natural number n? 97. In Exercise 6, we has you verify an identity which expresses sin(3 as a polynomial in terms of sin(. Can you do the same for sin(? What about for sin(? If not, what goes wrong? 98. Verify the Even / Odd Identities for tangent, secant, cosecant and cotangent. 99. Verify the Cofunction Identities for tangent, secant, cosecant and cotangent. 00. Verify the Difference Identities for sine and tangent. 0. Verify the Product to Sum Identities. 0. Verify the Sum to Product Identities.
18 0. Trigonometric Identities Answers 6 7. cos( sin(0 8. sec( csc(9 6 ( + 6. cot( 3. cos tan( ( 3π 6 +. sin ( π ( 3π. tan ( 7π cos ( 7π 7. tan + ( π sin ( π 9. cot ( + ( π 3 0. csc 6 (. sec π 6. (a cos(α + β (c tan(α + β 7 0 (b sin(α + β 7 0 (d cos(α β (e sin(α β (f tan(α β 3. (a cos(α + β (c tan(α + β (b sin(α + β 8 30 (d cos(α β (e sin(α β (a sin(α + β 6 6 (b cos(α β 33 6 (f tan(α β (c tan(α β 6 33
19 788 Foundations of Trigonometry. (a csc(α β (b sec(α + β 7 (c cot(α + β cos( sin( cos( tan(. +. cos. sin(7. ( 7π 3 ( π 3. sin ( π + 7. sin cos 8 8. tan ( 7π sin( sin ( 0 0. sin( sin ( sin( 0 69 sin ( sin( 8 sin ( sin( sin ( cos( 7 6 cos ( 7 0 cos( 809 cos ( cos( 9 69 cos ( 3 3 cos( 7 8 cos ( 8 cos( 7 cos ( tan( tan ( 7 tan( 0 tan ( 9 tan( 0 9 tan ( 3 tan( 7 tan ( tan ( + tan( 7 tan (
20 0. Trigonometric Identities 789. sin( sin (. sin( 0 69 sin ( sin( 0 69 sin ( 6 6 cos( 7 cos ( cos( 9 69 cos ( 6 6 cos( 9 69 cos ( 6 6 tan( 7 tan ( tan( 0 9 tan ( tan( 0 9 tan ( 7. sin( sin ( cos( 3 cos ( tan( 3 tan ( + tan ( 0 8. sin( sin ( cos( 3 cos ( tan( 3 tan ( + tan ( cos( + cos(8 cos( + cos(8 80. cos( cos( cos( cos(9 cos( cos( ( 9 8. cos 83. cos( sin( 8. cos ( π ( sin 76. sin(8 + sin(0 sin( + sin( 79. ( ( 8. sin sin 8. ( sin π 90. x 9. x x ln x + x + 6 ln(
3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότεραPractice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π 2, π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότερα10/3/ revolution = 360 = 2 π radians = = x. 2π = x = 360 = : Measures of Angles and Rotations
//.: Measures of Angles and Rotations I. Vocabulary A A. Angle the union of two rays with a common endpoint B. BA and BC C. B is the vertex. B C D. You can think of BA as the rotation of (clockwise) with
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραPARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Διαβάστε περισσότεραTrigonometric Formula Sheet
Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραF-TF Sum and Difference angle
F-TF Sum and Difference angle formulas Alignments to Content Standards: F-TF.C.9 Task In this task, you will show how all of the sum and difference angle formulas can be derived from a single formula when
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότερα2 2 2 The correct formula for the cosine of the sum of two angles is given by the following theorem.
5 TRIGONOMETRIC FORMULAS FOR SUMS AND DIFFERENCES The fundamental trignmetric identities cnsidered earlier express relatinships amng trignmetric functins f a single variable In this sectin we develp trignmetric
Διαβάστε περισσότερα2 2 2 The correct formula for the cosine of the sum of two angles is given by the following theorem.
5 TRIGONOMETRIC FORMULAS FOR SUMS AND DIFFERENCES The fundamental trignmetric identities cnsidered earlier express relatinships amng trignmetric functins f a single variable In this sectin we develp trignmetric
Διαβάστε περισσότεραDerivations of Useful Trigonometric Identities
Derivations of Useful Trigonometric Identities Pythagorean Identity This is a basic and very useful relationship which comes directly from the definition of the trigonometric ratios of sine and cosine
Διαβάστε περισσότερα9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr
9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values
Διαβάστε περισσότεραSection 7.7 Product-to-Sum and Sum-to-Product Formulas
Section 7.7 Product-to-Sum and Sum-to-Product Fmulas Objective 1: Express Products as Sums To derive the Product-to-Sum Fmulas will begin by writing down the difference and sum fmulas of the cosine function:
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραSection 9.2 Polar Equations and Graphs
180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότερα(a,b) Let s review the general definitions of trig functions first. (See back cover of your book) sin θ = b/r cos θ = a/r tan θ = b/a, a 0
TRIGONOMETRIC IDENTITIES (a,b) Let s eview the geneal definitions of tig functions fist. (See back cove of you book) θ b/ θ a/ tan θ b/a, a 0 θ csc θ /b, b 0 sec θ /a, a 0 cot θ a/b, b 0 By doing some
Διαβάστε περισσότεραReview Test 3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. 1) sin - 11π 1 1) + - + - - ) sin 11π 1 ) ( -
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραTrigonometry 1.TRIGONOMETRIC RATIOS
Trigonometry.TRIGONOMETRIC RATIOS. If a ray OP makes an angle with the positive direction of X-axis then y x i) Sin ii) cos r r iii) tan x y (x 0) iv) cot y x (y 0) y P v) sec x r (x 0) vi) cosec y r (y
Διαβάστε περισσότεραPg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is
Pg. 9. The perimeter is P = The area of a triangle is A = bh where b is the base, h is the height 0 h= btan 60 = b = b In our case b =, then the area is A = = 0. By Pythagorean theorem a + a = d a a =
Διαβάστε περισσότεραMATH 150 Pre-Calculus
MATH 150 Pre-Calculus Fall, 014, WEEK 11 JoungDong Kim Week 11: 8A, 8B, 8C, 8D Chapter 8. Trigonometry Chapter 8A. Angles and Circles The size of an angle may be measured in revolutions (rev), in degree
Διαβάστε περισσότεραTRIGONOMETRIC FUNCTIONS
Chapter TRIGONOMETRIC FUNCTIONS. Overview.. The word trigonometry is derived from the Greek words trigon and metron which means measuring the sides of a triangle. An angle is the amount of rotation of
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραLecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
Διαβάστε περισσότεραProblem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Διαβάστε περισσότεραSimilarly, we may define hyperbolic functions cosh α and sinh α from the unit hyperbola
Universit of Hperbolic Functions The trigonometric functions cos α an cos α are efine using the unit circle + b measuring the istance α in the counter-clockwise irection along the circumference of the
Διαβάστε περισσότερα( ) 2 and compare to M.
Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραMathCity.org Merging man and maths
MathCity.org Merging man and maths Exercise 10. (s) Page Textbook of Algebra and Trigonometry for Class XI Available online @, Version:.0 Question # 1 Find the values of sin, and tan when: 1 π (i) (ii)
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραQuadratic Expressions
Quadratic Expressions. The standard form of a quadratic equation is ax + bx + c = 0 where a, b, c R and a 0. The roots of ax + bx + c = 0 are b ± b a 4ac. 3. For the equation ax +bx+c = 0, sum of the roots
Διαβάστε περισσότεραProblem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.
Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +
Διαβάστε περισσότεραis like multiplying by the conversion factor of. Dividing by 2π gives you the
Chapter Graphs of Trigonometric Functions Answer Ke. Radian Measure Answers. π. π. π. π. 7π. π 7. 70 8. 9. 0 0. 0. 00. 80. Multipling b π π is like multipling b the conversion factor of. Dividing b 0 gives
Διαβάστε περισσότεραforms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with
Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραb. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότεραMath 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραCHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραChapter 7 Analytic Trigonometry
Chapter 7 Analytic Trigonometry Section 7.. Domain: { is any real number} ; Range: { y y }. { } or { }. [, ). True. ;. ; 7. sin y 8. 0 9. 0. False. The domain of. True. True.. y sin is. sin 0 We are finding
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραSOLVING CUBICS AND QUARTICS BY RADICALS
SOLVING CUBICS AND QUARTICS BY RADICALS The purpose of this handout is to record the classical formulas expressing the roots of degree three and degree four polynomials in terms of radicals. We begin with
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραSecond Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Διαβάστε περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότεραCHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS
CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS EXERCISE 01 Page 545 1. Use matrices to solve: 3x + 4y x + 5y + 7 3x + 4y x + 5y 7 Hence, 3 4 x 0 5 y 7 The inverse of 3 4 5 is: 1 5 4 1 5 4 15 8 3
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραNowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραCHAPTER 12: PERIMETER, AREA, CIRCUMFERENCE, AND 12.1 INTRODUCTION TO GEOMETRIC 12.2 PERIMETER: SQUARES, RECTANGLES,
CHAPTER : PERIMETER, AREA, CIRCUMFERENCE, AND SIGNED FRACTIONS. INTRODUCTION TO GEOMETRIC MEASUREMENTS p. -3. PERIMETER: SQUARES, RECTANGLES, TRIANGLES p. 4-5.3 AREA: SQUARES, RECTANGLES, TRIANGLES p.
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραOn a four-dimensional hyperbolic manifold with finite volume
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In
Διαβάστε περισσότεραSolution to Review Problems for Midterm III
Solution to Review Problems for Mierm III Mierm III: Friday, November 19 in class Topics:.8-.11, 4.1,4. 1. Find the derivative of the following functions and simplify your answers. (a) x(ln(4x)) +ln(5
Διαβάστε περισσότεραIntegrals in cylindrical, spherical coordinates (Sect. 15.7)
Integrals in clindrical, spherical coordinates (Sect. 5.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραFractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότεραHISTOGRAMS AND PERCENTILES What is the 25 th percentile of a histogram? What is the 50 th percentile for the cigarette histogram?
HISTOGRAMS AND PERCENTILES What is the 25 th percentile of a histogram? The point on the horizontal axis such that of the area under the histogram lies to the left of that point (and to the right) What
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραTMA4115 Matematikk 3
TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραLecture 13 - Root Space Decomposition II
Lecture 13 - Root Space Decomposition II October 18, 2012 1 Review First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra h (which we are calling a CSA, or Cartan Subalgebra).
Διαβάστε περισσότεραw o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότεραNotes on the Open Economy
Notes on the Open Econom Ben J. Heijdra Universit of Groningen April 24 Introduction In this note we stud the two-countr model of Table.4 in more detail. restated here for convenience. The model is Table.4.
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότερα5.4 The Poisson Distribution.
The worst thing you can do about a situation is nothing. Sr. O Shea Jackson 5.4 The Poisson Distribution. Description of the Poisson Distribution Discrete probability distribution. The random variable
Διαβάστε περισσότεραEQUATIONS OF DEGREE 3 AND 4.
EQUATIONS OF DEGREE AND 4. IAN KIMING Consider the equation. Equations of degree. x + ax 2 + bx + c = 0, with a, b, c R. Substituting y := x + a, we find for y an equation of the form: ( ) y + py + 2q
Διαβάστε περισσότερα