6. Determine whether the function below is or is not periodic. If it is, identify one cycle in two different ways and find the period and amplitude. 7. Sketch the graph of a wave with a period of 2 and an amplitude of 4. 8. Sketch the graph of a wave with a period of 4 and an amplitude of 3. 13-2 Angles and the Unit Circle Quick Review An angle is in standard position if the vertex is at the origin and one ray, the initial side, is on the positive x-axis. The other ray is the terminal side of the angle. Two angles in standard position are coterminal if they have the same terminal side. The unit circle has radius of 1 unit and its center at the origin. The cosine of e (cos @) is the x-coordinate of the point where the terminal side of the angle intersects the unit circle. The sine of 8 (sin 0) is the y-coordinate. Example What are the cosine and sine of -210*? Sketch an angle of -210* in standard position with a unit circle. The terminal side forms a 30 -60 - 90 triangle with hypotenuse = 1, shorter leg = _ ,longer leg - Y Since the terminal side lies in Quadrant II, cos( -210") is negative and sin(-210*) is positive cos (-210') - V3 , and sin (-2109) - Exercises 9. Find the measurement of the angle in standard position below. 10. Sketch a -30* angle in standard position. 11. Find the measure of an angle between 0" and 360 coterminal with a - 120* angle. 12. Find the exact values of the sine and cosine of 315* and -315. Then find the decimal equivalents. Round your answers to the nearest hundredth.13-3 Radian Measure Quick Review A central angle of a circle is an angle whose vertex is at the center of a circle and whose sides are radii of the circle. An intercepted arc is the portion of the circle whose endpoints are on the sides of the angle and whose remaining points lie in the interior of the angle. A radian is the measure of a central angle that intercepts an arc equal in length to a radius of the circle. Example What is the radian measure of an angle of - 210*? -210 - -2103 1805 radians 6 radians Exercises The measure e of an angle in standard position is given. a. Write each degree measure in radians and each radian measure in degrees rounded to the nearest degree. b. Find the exact values of cos 0 and sin 0 for each angle measure. 13. 60' 14. -45 15. 180 16. 2TT radians 17. radians 18. - 37 radians 19. Use the circle to find the length of the indicated arc. Round your answer to the nearest tenth. 13-4 The Sine Function Quick Review The sine function y = sin 0 matches the measure @ of an angle in standard position with the y-coordinate of a point on the unit circle. This point is where the terminal side of the angle intersects the unit circle, The graph of a sine function is called a sine curve. For the sine function y = a sin be, the amplitude equals a, there are b cycles from 0 to 2Tr, and the period is -I Example Determine the number of cycles the sine function y - - 7 sis30 has in the interval from 0 to 217. Find the amplitude and period of each function. For y = - 7 sin 30, a = -7 and b - 3. Therefore there are 3 cycles from 0 to 2Ti. The amplitude is a| - | -7 - 7. The period is 2 Exercises Sketch the graph of each function in the interval from 0 to 2Tr 20. y = 3 sin 0 21. y - sin 40 22. Write an equation of a sine function with a > 0, amplitude 4, and period 0.5TT13-5 The Cosine Function Quick Review The cosine function y = cos 8 matches the measure @ of an angle in standard position with the x-coordinate of a point on the unit circle. This point is where the terminal side of the angle intersects the unit circle. For the cosine function y = a cos be, the amplitude equals | a |, there are b cycles from 0 to 2Tr, and the period is " Example Find all solutions to 5 cos 8 - -4 in the interval from 0 to 217. Round each answer to the nearest hundredth. On a graphing calculator graph the equations y -4 and y - 5 cos 8. Use the Intersect feature to find the points at which the two graphs intersect. The graph shows two solutions in the interval. They are 0 ~ 2. 50 and 3. 79 Exercises Sketch the graph of each function in the interval from 0 to 2TT. 23. y = 2 cos (5 0) 24. y = - cos 20 25. Write an equation of a cosine function with a > 0, amplitude 3, and period TT. Solve each equation in the interval from 0 to 2Tr. Round your answer to the nearest hundredth. 26. 3 cos 40 27. COs (TO) - -0.6 13-6 The Tangent Function Quick Review The tangent of an angle 9 in standard position is the y-coordinate of the point where the terminal side of the angle intersects the tangent line z = 1. A tangent function in the form y = a tan be has a period of -. Example What is the period of y = tan -0? Tell where two asymptotes occur, period One cycle occurs in the interval from -2 to 2, so there are asymptotes at @ 2 and 8 - 2. Exercises Graph each function in the interval from 0 to 21. Then evaluate the function at t and t = 5. If the tangent is undefined at that point, write undefined. 28. y = tan =t 29. y = tan 3t 30. y - 2 tant 31. y = 4 tan 213-7 Translating Sine and Cosine Functions Quick Review Each horizontal translation of certain periodic functions is a phase shift. When g(z ) - f(z - h) + k, the value of p is the amount of the horizontal shift and the value of k is the amount of the vertical shift. Example What is an equation for the translation of y = sina, 2 units to the right and 1 unit up? 2 units to the right means h = 2 and 1 unit up means /: An equation is y = sin(z - 2) + 1. Exercises Graph each function in the inteval from 0 to 2TT. 32. y = cos (z + 5 ) 63. y = 2 sin z - 4 34. y = sin(1 - 7) + 3 35. y = Cos(1 + 7) - 1 Write an equation for each translation. 36. y = sin z, * units to the right 37. y = cos z, 2 units down 13-8 Reciprocal Trigonometric Functions Quick Review The cosecant (csc), secant (sec), and cotangent (cot) functions are defined as reciprocals for all real numbers 8 except those that make a denominator zero). 1 csc 8 = sec @ cote sin e cos @ an e Example Suppose sin 0 - - . Find csc 0. csco sin # Exercises Evaluate each expression. Write your answer in exact form. 38. sec(-45.) 39. cot 120 40. csc 150 41. cot (-150) Graph each function in the interval from 0 to 4TT. 42. y - 2csco 43. y - sec 0 - 1 44. y = cot - 0 45. y = csc = 0 + 2