Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4-2) Then/Now New Vocabulary Key Concept: Trigonometric Functions of Any Angle Example 1: Evaluate.

Presentation on theme: "Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4-2) Then/Now New Vocabulary Key Concept: Trigonometric Functions of Any Angle Example 1: Evaluate."— Presentation transcript:

1 Splash Screen

2 Lesson Menu 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

3 Over Lesson 4-2 5–Minute Check 1 Write 62.937˚ in DMS form. A.62°54'13" B.63°22'2" C.62°54'2" D.62°56'13.2"

4 Over Lesson 4-2 5–Minute Check 2 Write 96°42'16'' in decimal degree form to the nearest thousandth. A.96.704 o B.96.422 o C.96.348 o D.96.259 o

5 Over Lesson 4-2 5–Minute Check 3 Write 135º in radians as a multiple of π. A. B. C. D.

6 Over Lesson 4-2 5–Minute Check 4 A.240 o B.–60 o C.–120 o D.–240 o Write in degrees.

7 Over Lesson 4-2 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

8 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.

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

10 Key Concept 1

11 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.

12 Example 1 Evaluate Trigonometric Functions Given a Point Answer:

13 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.

14 Key Concept 2

15 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.

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

17 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.

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

19 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.

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

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

22 Key Concept 3

23 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°

24 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.

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

26 Key Concept 4

27 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

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

29 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.

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

31 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.

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

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

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

35 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.

36 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

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

38 Example 5 Use One Trigonometric Value to Find Others Answer:

39 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.

40 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

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

42 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

43 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.

44 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

45 Key Concept 7

46 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

47 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.

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

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

50 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°

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

52 Key Concept 8

53 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.

54 Example 8 Use the Periodic Nature of Circular Functions Answer:

55 Example 8 B. Find the exact value of sin(–300). sin (–300 o ) = sin (60 o + 360 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 + 360 o (–1) map to the same point (x, y) = on the unit circle.

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

57 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.

58 Example 8 Use the Periodic Nature of Circular Functions Answer:

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

60 End of the Lesson