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Five-Minute Check (over Lesson 4-2) Then/Now New Vocabulary Key Concept: Trigonometric Functions of Any Angle Example 1: Evaluate Trigonometric Functions Given a Point Key Concept: Common Quadrantal Angles Example 2: Evaluate Trigonometric Functions of Quadrantal Angles Key Concept: Reference Angle Rules Example 3: Find Reference Angles Key Concept: Evaluating Trigonometric Functions of Any Angle Example 4: Use Reference Angles to Find Trigonometric Values Example 5: Use One Trigonometric Value to Find Others Example 6: Real-World Example: Find Coordinates Given a Radius and an Angle Key Concept: Trigonometric Functions on the Unit Circle Example 7: Find Trigonometric Values Using the Unit Circle Key Concept: Periodic Functions Example 8: Use the Periodic Nature of Circular Functions Lesson Menu

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

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 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 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

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

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

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

Take the positive square root. 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. Pythagorean Theorem x = –4 and y = 3 Take the positive square root. Use x = –4, y = 3, and r = 5 to write the six trigonometric ratios. Example 1

Evaluate Trigonometric Functions Given a Point Answer: Example 1

Evaluate Trigonometric Functions Given a Point Answer: Example 1

Let (–3, 6) be a point on the terminal side of an angle Ө in standard position. Find the exact values of the six trigonometric functions of Ө. A. B. C. D. Example 1

Let (–3, 6) be a point on the terminal side of an angle Ө in standard position. Find the exact values of the six trigonometric functions of Ө. A. B. C. D. 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. The terminal side of π in standard position lies on the negative x-axis. Choose a point P on the terminal side of the angle. A convenient point is (–1, 0) because r = 1. Example 2

Cosine function x = –1 and r = 1 Answer: Evaluate Trigonometric Functions of Quadrantal Angles Cosine function x = –1 and r = 1 Answer: Example 2

Cosine function x = –1 and r = 1 Answer: –1 Evaluate Trigonometric Functions of Quadrantal Angles Cosine function x = –1 and r = 1 Answer: –1 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. The terminal side of 450° in standard position lies on the positive y-axis. Choose a point P(0, 1) on the terminal side of the angle because r = 1. Example 2

Tangent function y = 1 and x = 0 Answer: Evaluate Trigonometric Functions of Quadrantal Angles Tangent function y = 1 and x = 0 Answer: Example 2

Tangent function y = 1 and x = 0 Answer: undefined Evaluate Trigonometric Functions of Quadrantal Angles Tangent function y = 1 and x = 0 Answer: 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. The terminal side of in standard position lies on the negative y-axis. The point (0, –1) is convenient because r = 1. Example 2

Cotangent function x = 0 and y = –1 Answer: Evaluate Trigonometric Functions of Quadrantal Angles Cotangent function x = 0 and y = –1 Answer: Example 2

Cotangent function x = 0 and y = –1 Answer: 0 Evaluate Trigonometric Functions of Quadrantal Angles Cotangent function x = 0 and y = –1 Answer: 0 Example 2

Find the exact value of sec If not defined, write undefined. B. 0 C. 1 D. undefined Example 2

Find the exact value of sec If not defined, write undefined. B. 0 C. 1 D. 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. A coterminal angle is –150° + 360° or 210°. The terminal side of 210° lies in Quadrant III. Therefore, its reference angle is 210° – 180° or 30°. Answer: Example 3

A. Sketch –150°. Then find its reference angle. Find Reference Angles A. Sketch –150°. Then find its reference angle. A coterminal angle is –150° + 360° or 210°. The terminal side of 210° lies in Quadrant III. Therefore, its reference angle is 210° – 180° or 30°. Answer: 30° Example 3

B. Sketch . Then find its reference angle. Find Reference Angles B. Sketch . Then find its reference angle. The terminal side of lies in Quadrant II. Therefore, its reference angle is . Answer: Example 3

B. Sketch . Then find its reference angle. Find Reference Angles B. Sketch . Then find its reference angle. The terminal side of lies in Quadrant II. Therefore, its reference angle is . Answer: Example 3

Find the reference angle for a 520o angle. B. 70° C. 160° D. 200° Example 3

Find the reference angle for a 520o angle. B. 70° C. 160° D. 200° Example 3

Key Concept 4

A. Find the exact value of . Use Reference Angles to Find Trigonometric Values A. Find the exact value of . Because the terminal side of  lies in Quadrant III, the reference angle Example 4

In Quadrant III, sin θ is negative. Use Reference Angles to Find Trigonometric Values In Quadrant III, sin θ is negative. Answer: Example 4

In Quadrant III, sin θ is negative. Use Reference Angles to Find Trigonometric Values In Quadrant III, sin θ is negative. Answer: 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. Because the terminal side of θ lies in Quadrant II, the reference angle θ' is 180o – 150o or 30o. Example 4

tan 150° = –tan 30° In Quadrant II, tan θ is negative. Use Reference Angles to Find Trigonometric Values tan 150° = –tan 30° In Quadrant II, tan θ is negative. tan 30° Answer: Example 4

tan 150° = –tan 30° In Quadrant II, tan θ is negative. Use Reference Angles to Find Trigonometric Values tan 150° = –tan 30° In Quadrant II, tan θ is negative. tan 30° Answer: Example 4

C. Find the exact value of . Use Reference Angles to Find Trigonometric Values C. Find the exact value of . A coterminal angle of which lies in Quadrant IV. So, the reference angle Because cosine and secant are reciprocal functions and cos θ is positive in Quadrant IV, it follows that sec θ is also positive in Quadrant IV. Example 4

In Quadrant IV, sec θ is positive. Use Reference Angles to Find Trigonometric Values In Quadrant IV, sec θ is positive. Example 4

Use Reference Angles to Find Trigonometric Values Answer: Example 4

CHECK You can check your answer by using a graphing calculator. Use Reference Angles to Find Trigonometric Values Answer: CHECK You can check your answer by using a graphing calculator. Example 4

Find the exact value of cos . B. C. D. Example 4

Find the exact value of cos . B. C. D. 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. To find the other function values, you must find the coordinates of a point on the terminal side of θ. You know that sec θ is positive and sin θ is positive, so θ must lie in Quadrant I. This means that both x and y are positive. Example 5

Because sec  = and x = 5 to find y. Use One Trigonometric Value to Find Others Because sec  = and x = 5 to find y. Pythagorean Theorem r = and x = 5 Take the positive square root. Example 5

Use One Trigonometric Value to Find Others Use x = 5, y = 2, and r = to write the other five trigonometric ratios. Example 5

Use One Trigonometric Value to Find Others Answer: Example 5

Use One Trigonometric Value to Find Others Answer: Example 5

Let csc θ = –3, tan θ < 0. Find the exact values o the five remaining trigonometric functions of θ. A. B. C. D. Example 5

Let csc θ = –3, tan θ < 0. Find the exact values o the five remaining trigonometric functions of θ. A. B. C. D. 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

Cosine ratio  = 150° and r = 10 cos 150° = –cos 30° Solve for x. Find Coordinates Given a Radius and an Angle Cosine ratio  = 150° and r = 10 cos 150° = –cos 30° Solve for x. Example 6

Sin ratio θ = 150° and r = 10 sin 150° = sin 30° 5 = y Solve for y. Find Coordinates Given a Radius and an Angle Sin ratio θ = 150° and r = 10 sin 150° = sin 30° Solve for y. 5 = y Example 6

Find Coordinates Given a Radius and an Angle Answer: 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

A. 6 feet left and 3.5 feet above the pivot point CLOCK TOWER A 4-foot long minute hand on a clock on a bell tower shows a time of 15 minutes past the hour. What is the new position of the end of the minute hand relative to the pivot point at 5 minutes before the next hour? A. 6 feet left and 3.5 feet above the pivot point B. 3.4 feet left and 2 feet above the pivot point C. 3.4 feet left and 6 feet above the pivot point D. 2 feet left and 3.5 feet above the pivot point Example 6

A. 6 feet left and 3.5 feet above the pivot point CLOCK TOWER A 4-foot long minute hand on a clock on a bell tower shows a time of 15 minutes past the hour. What is the new position of the end of the minute hand relative to the pivot point at 5 minutes before the next hour? A. 6 feet left and 3.5 feet above the pivot point B. 3.4 feet left and 2 feet above the pivot point C. 3.4 feet left and 6 feet above the pivot point D. 2 feet left and 3.5 feet 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. corresponds to the point (x, y) = on the unit circle. Definition of sin t sin t = y y = . sin Answer: Example 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. corresponds to the point (x, y) = on the unit circle. Definition of sin t sin t = y y = . sin Answer: 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. corresponds to the point (x, y) = on the unit circle. cos t = x Definition of cos t cos Answer: 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. corresponds to the point (x, y) = on the unit circle. cos t = x Definition of cos t cos Answer: Example 7

C. Find the exact value of . If undefined, write undefined. Find Trigonometric Values Using the Unit Circle C. Find the exact value of . If undefined, write undefined. Definition of tan t. Example 7

Simplify. Answer: Find Trigonometric Values Using the Unit Circle Example 7

Simplify. Answer: Find Trigonometric Values Using the Unit Circle Example 7

D. Find the exact value of sec 270°. If undefined, write undefined. Find Trigonometric Values Using the Unit Circle D. Find the exact value of sec 270°. If undefined, write undefined. 270° corresponds to the point (x, y) = (0, –1) on the unit circle. Definition of sec t x = 0 when t = 270° Therefore, sec 270° is undefined. Answer: Example 7

D. Find the exact value of sec 270°. If undefined, write undefined. Find Trigonometric Values Using the Unit Circle D. Find the exact value of sec 270°. If undefined, write undefined. 270° corresponds to the point (x, y) = (0, –1) on the unit circle. Definition of sec t x = 0 when t = 270° Therefore, sec 270° is undefined. Answer: undefined Example 7

Find the exact value of tan . If undefined, write undefined. B. C. D. Example 7

Find the exact value of tan . If undefined, write undefined. B. C. D. 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

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

= 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

Use the Periodic Nature of Circular Functions Answer: Example 8

Find the exact value of cos B. –1 C. D. Example 8

Find the exact value of cos B. –1 C. D. Example 8

End of the Lesson