Then/Now 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.

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

Key Concept 1

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

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.

Key Concept 2

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

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

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

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

Key Concept 3

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

Example 3 Find the reference angle for a 520 o angle. A.20° B.70° C.160° D.200°

Key Concept 4

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

Example 4 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 180 o – 150 o or 30 o.

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

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

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

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

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

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

Example 6 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. Find Coordinates Given a Radius and an Angle

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

Example 6 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: 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 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

Key Concept 7

Example 7 Find Trigonometric Values Using the Unit Circle Definition of sin tsin t = y Answer: A. Find the exact value of. If undefined, write undefined. corresponds to the point (x, y) = on the unit circle. y =. sin

Example 7 Find Trigonometric Values Using the Unit Circle Answer: cos t= xDefinition of cos t cos corresponds to the point (x, y) = on the unit circle. B. Find the exact value of. If undefined, write undefined.

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

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

Example 7 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. Therefore, sec 270° is undefined. Answer: undefined Definition of sec t x = 0 when t = 270°

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

Key Concept 8

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

Example 8 Use the Periodic Nature of Circular Functions Answer:

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

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

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

End of the Lesson