Trigonometric Functions on the Unit Circle LESSON 4–3 Trigonometric Functions on the Unit Circle

Write 62.937˚ in DMS form. A. 62°54'13" B. 63°22'2" C. 62°54'2" 5–Minute Check 1

Write 96°42'16'' in decimal degree form to the nearest thousandth. A. 96.704o B. 96.422o C. 96.348o D. 96.259o 5–Minute Check 2

Write 135º in radians as a multiple of π. B. C. D. 5–Minute Check 3

Write in degrees. A. 240o B. –60o C. –120o D. –240o 5–Minute Check 4

Find the length of the intercepted arc with a central angle of 60° in a circle with a radius of 15 centimeters. Round to the nearest tenth. A. 7.9 cm B. 14.3 cm C. 15.7 cm D. 19.5 cm 5–Minute Check 5

Targeted TEKS P.4(A) Determine the relationship between the unit circle and the definition of a periodic function to evaluate trigonometric functions in mathematical and real-world problems. P.4(B) Describe the relationship between degree and radian measure on the unit circle. P.4(C) Represent angles in radians or degrees based on the concept of rotation and find the measure of reference angles and angles in standard position. Also addresses P.2(P), P.4(E), and P.4(F). Mathematical Processes P.1(D), P.1(E)

You found values of trigonometric functions for acute angles using ratios in right triangles. (Lesson 4-1) Find values of trigonometric functions for any angle. Find values of trigonometric functions using the unit circle. Then/Now

quadrantal angle reference angle unit circle circular function periodic function period Vocabulary

Key Concept 1

Find the exact values of the six trigonometric functions of θ. Evaluate Trigonometric Functions Given a Point Let (–4, 3) be a point on the terminal side of an angle θ in standard position. Find the exact values of the six trigonometric functions of θ. Example 1

Key Concept 2

A. Find the exact value of cos π. If not defined, write undefined. Evaluate Trigonometric Functions of Quadrantal Angles A. Find the exact value of cos π. If not defined, write undefined. Example 2

B. Find the exact value of tan 450°. If not defined, write undefined. Evaluate Trigonometric Functions of Quadrantal Angles B. Find the exact value of tan 450°. If not defined, write undefined. Example 2

C. Find the exact value of . If not defined, write undefined. Evaluate Trigonometric Functions of Quadrantal Angles C. Find the exact value of . If not defined, write undefined. Example 2

Key Concept 3

A. Sketch –150°. Then find its reference angle. Find Reference Angles A. Sketch –150°. Then find its reference angle. Example 3

B. Sketch . Then find its reference angle. Find Reference Angles B. Sketch . Then find its reference angle. Example 3

A. Find the exact value of . Use Reference Angles to Find Trigonometric Values A. Find the exact value of . Example 4

B. Find the exact value of tan 150º. Use Reference Angles to Find Trigonometric Values B. Find the exact value of tan 150º. Example 4

C. Find the exact value of . Use Reference Angles to Find Trigonometric Values C. Find the exact value of . Example 4

Use One Trigonometric Value to Find Others Let , where sin θ > 0. Find the exact values of the remaining five trigonometric functions of θ. Example 5

Use One Trigonometric Value to Find Others Example 5

Find Coordinates Given a Radius and an Angle ROBOTICS A student programmed a 10-inch long robotic arm to pick up an object at point C and rotate through an angle of 150° in order to release it into a container at point D. Find the position of the object at point D, relative to the pivot point O. Example 6

Find Coordinates Given a Radius and an Angle Example 6

Find Coordinates Given a Radius and an Angle Answer: The exact coordinates of D are . The object is about 8.66 inches to the left of the pivot point and 5 inches above the pivot point. Example 6

Key Concept 7

A. Find the exact value of . If undefined, write undefined. Find Trigonometric Values Using the Unit Circle A. Find the exact value of . If undefined, write undefined. sin t = y sin Example 7

B. Find the exact value of . If undefined, write undefined. Find Trigonometric Values Using the Unit Circle B. Find the exact value of . If undefined, write undefined. cos t = x cos Example 7

Key Concept 8

A. Find the exact value of . Use the Periodic Nature of Circular Functions A. Find the exact value of . Rewrite as the sum of a number and 2π. + 2π map to the same point (x, y) = on the unit circle. cos t = x and x = Example 8

Use the Periodic Nature of Circular Functions Answer: Example 8

B. Find the exact value of sin(–300). Use the Periodic Nature of Circular Functions B. Find the exact value of sin(–300). sin (–300o) = sin (60o + 360o(–1)) Rewrite –300o as the sum of a number and an integer multiple of 360o. = sin 60o 60o and 60o + 360o(–1) map to the same point (x, y) = on the unit circle. Example 8

= sin t = y and y = when t = 60o. Answer: Use the Periodic Nature of Circular Functions = sin t = y and y = when t = 60o. Answer: Example 8

C. Find the exact value of . Use the Periodic Nature of Circular Functions C. Find the exact value of . Rewrite as the sum of a number and 2 and an integer multiple of π. map to the same point (x, y) = on the unit circle. Example 8

Use the Periodic Nature of Circular Functions Answer: Example 8

Trigonometric Functions on the Unit Circle LESSON 4–3 Trigonometric Functions on the Unit Circle