10/3/ revolution = 360 = 2 π radians = = x. 2π = x = 360 = : Measures of Angles and Rotations

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1 //.: 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 center B. BC E. Counter-Clockwise rotations are positive F. Clockwise rotations are negative G. Rotations can be measured in terms of revolutions II. Revolutions and Degrees A. counterclockwise revolution is degrees B. Example : Convert to degrees. / revolution counterclockwise x x. / revolution clockwise x x. / revolution clockwise x. / revolutions counterclockwise x III. Radians and Degrees A. Radians another way to measure angles. degrees radians. (You are measuring circumference traveled along a circle with radius a unit circle.) C r () C revolution radians B. Example : Convert to radians. Give exact values. ½ revolution counterclockwise x. / revolution clockwise x. / revolution counterclockwise x

2 //. / revolution clockwise x. / revolution clockwise x. / revolutions counterclockwise x IV. Conversion from Radians to Degrees A.,9 formula radians degrees,,,7, B. Example : Convert to radians. Give both exact and approximate values (hundredth).. x x.. 9 x 9 9 x C. Example : How many revolutions equal radians (approx)?.7 rev x D. How many revolutions equal radians 7 7 x 7 Homework: p. - #,,-9, -,, 7, 9,, -.

3 // I. Finding Arc Length A. Example : Find the length of the arc of a circle with radius inches and central angle note: we have / of the entire circle s circumference.: LENGTHS OF ARCS AND AREAS OF SECTORS s C s ( ) s. in s total B. Example : Find the length of the arc of a circle with radius inches and central angle s C s s. in s total C. Circular Arc Length Formula: s arc length, measure of central angle (radians), r radius. Then in radians: s r in degrees: s ( r) total D. Example : Find the radius of a circle if the length of an arc is and its central angle is. s r r r r r total II. Finding Sector Area A. Find the area of a sector of a circle with radius note: we have / of the entire circle s area A A A ( Area of circle ) ( ) in total

4 // B. Circular Sector Area Formula: A area of the sector, measure of central angle (radians), r radius. Then in radians: A r in degrees: s r 7 total C. EX : Repeat example using radian measure and the formula above. A r A in A total III. Real-World example EX : A water irrigation arm meters long rotates around pivot P once per day. How much area is irrigated every hour? Include a picture. Homework: p. - # -7, 9-, -. m day day hours A ( ) A,7 m 9 total total

5 // I. Trigonometry on the Unit Circle A. The sine and cosine of an angle are defined as the coordinates of a point on the unit circle with an angle from the positive x-axis.,9 P (cos,sin),, B. Tangent is defined as follows:,7 sin tan, for cos cos C. Use the unit circle to complete the following: (, ),7 cos cos cos cos sin sin sin sin tan tan tan tan,9,, (, ) (, ) (, ) D. Example : Find the image of (,) under the rotation of. (Put your calculator in RADIAN mode.) 9, cos,sin 9 9 ( xy) ( xy, ) (.9,.) ( cos( ),sin( 9 )) II. Signs of Trig Functions Sin (y) Cos (x) Tan (y/x) < < < < < < < < I II III IV ,9,,,7 Students sine is positive,9 All,, Take tangent is positive - - -,7 sine, cosine, and tangent are positive Calculus cosine is positive

6 // III. Example : A large clock with a rotating hour and minute hand is on a building with center feet tall. The length of the minute hand is feet. At :, how far off the ground is the tip of the minute hand? y sin ( ) y ft height 9 ft y IX X VIII XI VII XII VI I V II IV III HW: p. - # -,

7 //.: Exact Values of Sines, Cosines, and Tangents I. Special Right Triangles A. First, the --9 triangle. B. If we place this triangle on the unit circle: s s... s s s, s. C. Next, the --9 triangle. D. If we place this triangle on the unit circle s s s s s..,..., s... E. All angles we know in the first quadrant look like this:, ( ),, II. Using the unit circle: A. Evaluate each of the following,, cos cos sin sin, tan tan an tan

8 //, ( ), cos cos sin sin, tan tan tan tan, ( ), cos cos, sin sin tan tan an tan III. The Entire Unit Circle Homework: p. - #-

9 // I. Definitions Three other trigonometric functions: cosecant: csc sin secant: sec cos cos cotangent: cot sin tan II. Examples A. Example : Calculate exact values for each of the following:.csc csc sin.csc sin.sec cos.cot cos sin.sec sec cos.cot cot tan B. You will notice that the trig functions take on the following values (they are paired into reciprocals): sin,cos,csc,sec : tan,cot : C. Example : Find all values of such that <.. sec. cot cos. csc 7 sin,. cot tan, tan. csc 7, sin,. sec cos 7,

10 // Homework: reciprocal function worksheet

11 // I. Pythagorean Identity: A. Note that for any point on the unit circle:,9 P (cos,sin) sin, cos ( ) ( ) sin + cos sin + cos B. Example : If, use sin the Pythagorean identity to find the value of. sin + cos + cos cos + 9 cos 9 cos cos ± 9 cos ± cos ± C. Example : If cos and <, < use a right triangle to find the cosine and tangent + y + y y - ± y y sin - tan y II. Other identities A. Opposites Theorem: sine cosine tangent B. Supplements Theorem: sine cosine tangent sin ( ) sin cos( ) cos tan ( ) tan sin ( ) sin cos ( ) cos tan ( ) tan

12 // C. Half Turn Theorem sine cosine tangent sin ( + ) sin cos ( + ) cos tan ( + ) tan + D. Properties of Complements For any right triangle b sin c b cos c sin cos c - a a cos c a sin c b cos sin E. Example : Given the following values for sine and cosine, find each of the following: sin, and cos sin. tan cos. cos( ) ( + ) cos( ). sin sin ( ) tan ( ) sin, and cos tan sin ( ). cos. sin ( ) sin Homework: p. #,,, 7, 9,,, 9, (a,b),, 7,