MATH 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 ( ) or in radians (rad). 1 rev = 360 = rad. 1

Note. To convert degrees to radians, multiply rad 180. To convert radians to degrees, multiply 180 rad.

Definition. An angle is called acute if its measure is between 0 and 90, and an angle is called obtuse if its measure is between 90 and 180. Definition. The complement of an angle whose measure is θ is any angle whose measure is 90 θ (in degrees), and the suppliment of an angle ψ is an angle whose measure is 180 ψ (in degrees). Definition. The circumference of the circle is the distance around the circle. By the definition of, the circumference is C = d = r, where d is diameter and r is radius. Definition. The area of the circle is A = r. Definition. An arc is any piece of the circle between two points on the circle. A chord is any line segment between two points on the circle. A sector is any piece of the disk between two radial lines. Arc length. Sector area. L = rθ for θ in radians. A = 1 rl = 1 r θ for θ in radians. 3

Ex1) Suppose the radius of circle is r = 6cm and the central angle is θ = 105. a) Find the circumference and area of the circle. b) Find the arc length and sector area by the central angle. 4

Definition. A secant line is a line which intersects the circle twice. The part inside the circle is a chord. Definition. A tangent line is a line which intersects the circle at exactly one point called the point of tangency. 5

Ex) A circular race track is being built. The outside diameter of the track is 00 ft and the inside diameter is 180 ft. If two runners run around the track once with one of them running on the outside edge of the track, while the other is running on the inside dege, how much further will the outside runner have to run? Ex3) If a circle has a 0 meter diameter, find the exact arc length and exact area of the sector subtended by a central angle of 15 degrees. 6

Chapter 8B. Trigonometric Functions and sinθ = opp hyp, cscθ = 1 sinθ = hyp opp cosθ = adj hyp, secθ = 1 cosθ = hyp adj tanθ = opp adj, cotθ = 1 tanθ = adj opp tanθ = sinθ cosθ The unit circle The unit circle is the circle of radius 1, centered at the origin in the xy-plane. Its equation is x +y = 1. 7

Note. Any right triangle has a similar right triangle in the unit circle. In unit circle, sinθ = y 1 = y cosθ = x 1 = x and tanθ = sinθ cosθ = y x = slope Special angles 0 6 4 3 3 sin θ 0 1 3 1 0-1 0 cosθ 1 3 1 0-1 0 1 tanθ 0 1 3 1 3-0 - 0 8

Signs of the Trigonometric Functions Reference Angle The smallest angle that the terminal side of a given angle makes with the x-axis. 9

Negative Angle Identities sin( θ) = sinθ cos( θ) = cosθ tan( θ) = tanθ Sumpplementary Angle Identities sin( θ) = sinθ cos( θ) = cosθ tan( θ) = tanθ sin( +θ) = sinθ cos( +θ) = cosθ tan( +θ) = tanθ 10

Ex4) Find the trig. functions at 3 4 rad. 11

Ex5) Find the trig. functions at 5 3 rad. 1

Ex6) Find the trig. functions at 3 rad. 13

Ex7) Find the point on the unit circle that corresponds to the angle θ = 5 6. Ex8) If cosθ = 0 1 and θ is in Quadrant IV, exactly find the trig. functions. 14

Chapter 8C. Graphs of the Trigonometric Functions Sine Graph y 1 3 x 1 Cosine Graph y 1 3 x 1 Tangent Graph y x 3 15

Cosecent y 1 1 3 x Secent y 1 1 3 x Cotangent y x 3 16

Periodic Properties The functions sine and cosine have period : sin(x+) = sinx cos(x+) = cosx The function tangent has period : tan(x+) = tanx Shifting and Rescaling Graphs The sine and cosine curves have amplitude a and period k. y = asinkx and y = acoskx (k > 0) An appropriate interval on which to graph one complete period is [ 0, k ]. Ex9) a) Graph y = sinx, y = sinx, and y = 4sinx. b) Graph y = sinx, y = sinx, y = sin 1 x, and y = sin 1 3 x. 17

Shifted Sine and Cosine Curves The sine and cosine curves y = asin(k(x b))+c and y = acos(k(x b))+c (k > 0) have amplitude a, period, phase shift b, and vertical shift c. k [ An appropriate interval on which to graph one complete period is b,b+ k ]. Ex10) Find the amplitude and period of each function and sketch its graph. a) y = 4cos3x b) y = sin 1 x 18

Ex11) Find the amplitude, period, and phase shift of y = sin(x ), and graph one complete 4 period. 19

Ex1) If f(x) = cos(3x )+1, what is the amplitude, period, phase shift, and vertical shift. And sketch the graph. 0

Ex13) A scientist recorded data from an experiment. She noticed that the data appeared to be periodic with period 4 and had a maximum value of 5, which occured at x = 3. After staring at the data for some time, she decided to use a cosine function to model the data. What was the form of the function she used. 1

Chapter 8D. Trigonometric Identities Pythagorean Identities sin θ+cos θ = 1 tan θ +1 = sec θ 1+cot θ = csc θ Negative Angle Identities sin( θ) = sinθ cos( θ) = cosθ tan( θ) = tanθ csc( θ) = cscθ sec( θ) = secθ cot( θ) = cotθ Note. Use the unit circle to understand.

Supplementary Angle Identities sin( θ) = sinθ cos( θ) = cosθ tan( θ) = tanθ csc( θ) = cscθ sec( θ) = secθ cot( θ) = cotθ Furthermore, sin( +θ) = sinθ cos( +θ) = cosθ tan( +θ) = tanθ csc( +θ) = cscθ sec( +θ) = secθ cot( +θ) = cotθ 3

Complementary Angle Identities ( ) sin θ ( ) θ csc ( ) = cosθ cos θ ( ) θ = secθ sec ( ) = sinθ tan θ ( ) θ = cscθ cot = cotθ = tanθ Furthermore, ( ) sin +θ ( ) ( ) = cosθ cos +θ = sinθ tan +θ = cotθ ( ) ( ) ( ) csc +θ = secθ sec +θ = cscθ cot +θ = tanθ 4

Sum and Difference of Two Angles Formulas sin(α+β) = sinαcosβ +cosαsinβ sin(α β) = sinαcosβ cosαsinβ cos(α+β) = cosαcosβ sinαsinβ cos(α β) = cosαcosβ +sinαsinβ tan(α+β) = tanα+tanβ 1 tanαtanβ tan(α β) = tanα tanβ 1+tanαtanβ Double Angle Formulas sin(α) = sinαcosα cos(α) = cos α sin α = cos α 1 = 1 sin α tan(α) = tanα 1 tan α 5

Ex14) If θ lies in the third quadrant, and cosθ = 1, what is the value of sinθ? 3 Ex15) Use a Sum or Difference Formulas to find the exact value of cos(75 ). 6

Ex16) Given that tanα = 1, where α isin Quadrant IIIandthat cotβ = 3 3, where β isin Quadrant II. Determine the exact values of the following. a) cscα b) sinβ c) cos(α β) 7

Ex17) Simplify the expression secθ tanθ tanθ secθ cotθ 8

Ex18) Verify tanxsinx = cscx cotx+cot 3 x 9

Ex19) Simplify sin x cos x+3 csc x cot x+ 30