LESSON 4–2 Degrees and Radians

Five-Minute Check (over Lesson 4-1) TEKS Then/Now New Vocabulary Example 1: Convert Between DMS and Decimal Degree Form Key Concept: Radian Measure Key Concept: Degree/Radian Conversion Rules Example 2: Convert Between Degree and Radian Measure Key Concept: Coterminal Angles Example 3: Find and Draw Coterminal Angles Key Concept: Arc Length Example 4: Find Arc Length Key Concept: Linear and Angular Speed Example 5: Real-World Example: Find Angular and Linear Speeds Key Concept: Area of a Sector Example 6: Find Areas of Sectors Lesson Menu

Find the exact values of the six trigonometric functions of θ. B. C. D. 5–Minute Check 1

If , find the exact values of the five remaining trigonometric functions of θ. B. C. D. 5–Minute Check 2

Solve ΔABC. Round side lengths to the nearest tenth and angle measures to the nearest degree. A. a ≈ 26.8, c ≈ 13.8, C = 31o B. a ≈ 19.7, c ≈ 11.8, C = 31o C. a ≈ 11.8, c ≈ 19.7, C = 31o D. a ≈ 15.1, c ≈ 17.3, C = 41o 5–Minute Check 3

Find the value of x. Round to the nearest tenth. B. 32.5 C. 15.7 D. 8.7 5–Minute Check 4

If , find tan θ. A. B. C. D. 5–Minute Check 5

Targeted TEKS P.2(P) Determine the values of the trigonometric functions at the special angles and relate them in mathematical and real-world problems. P.4(D) Represent angles in radians or degrees based on the concept of rotation in mathematical and real-world problems, including linear and angular velocity. Also addresses P.4(F). Mathematical Processes P.1(E), P.1(G)

Convert degree measures of angles to radian measures, and vice versa. You used the measures of acute angles in triangles given in degrees. (Lesson 4-1) Convert degree measures of angles to radian measures, and vice versa. Use angle measures to solve real-world problems. Then/Now

vertex initial side terminal side standard position radian coterminal angles linear speed angular speed sector Vocabulary

First, convert 0.125° into minutes and seconds. Convert Between DMS and Decimal Degree Form A. Write 329.125° in DMS form. First, convert 0.125° into minutes and seconds. 329.125° = 329° + 1° = 60' = 329° + 7.5' Simplify. Next, convert 0.5' into seconds. 329.125° = 329° + 7' + 1' = 60" = 329° + 7' + 30" Simplify. Therefore, 329.125° can be written as 329°7'30". Answer: 329°7'30" Example 1

B. Write 35°12'7'' in decimal degree form to the nearest thousandth. Convert Between DMS and Decimal Degree Form B. Write 35°12'7'' in decimal degree form to the nearest thousandth. Each minute is of a degree and each second is of a minute, so each second is of a degree. 35°12'7" = 35o + 12' Example 1

Therefore, 35°12'7" can be written as about 35.202°. Convert Between DMS and Decimal Degree Form ≈ 35° + 0.2 + 0.002 Simplify. ≈ 35.202° Add. Therefore, 35°12'7" can be written as about 35.202°. Answer: 35.202° Example 1

Write 141.275° in DMS form. A. 141°12'4.5" B. 141.2°45'0" C. 141°4'35" Example 1

Key Concept 2

Key Concept 2

A. Write 135° in radians as a multiple of π. Convert Between Degree and Radian Measure A. Write 135° in radians as a multiple of π. Answer: Example 2

B. Write –30° in radians as a multiple of π. Convert Between Degree and Radian Measure B. Write –30° in radians as a multiple of π. Answer: Example 2

C. Write in degrees. = 120° Simplify. Answer: 120° Convert Between Degree and Radian Measure C. Write in degrees. = 120° Simplify. Answer: 120° Example 2

D. Write in degrees. = 135° Simplify. Answer: –135° Convert Between Degree and Radian Measure D. Write in degrees. = 135° Simplify. Answer: –135° Example 2

Write 150o in radians as a multiple of π. B. C. D. Example 2

Key Concept 3

Find and Draw Coterminal Angles A. Identify all angles that are coterminal with 80°. Then find and draw one positive and one negative angle coterminal with 80°. All angles measuring 80° + 360n° are coterminal with an 80° angle. Let n = 1 and –1. 80° + 360(1)° = 80° + 360° or 440° Example 3

Answer: 80o + 360no; Sample answers: 440o, –280o Find and Draw Coterminal Angles 80° + 360(–1)° = 80° – 360° or –280° Answer: 80o + 360no; Sample answers: 440o, –280o Example 3

All angles measuring are coterminal with a angle. Let n = 1 and –1. Find and Draw Coterminal Angles B. Identify all angles that are coterminal with . Then find and draw one positive and one negative angle coterminal with . All angles measuring are coterminal with a angle. Let n = 1 and –1. Example 3

Answer: Sample answer: Find and Draw Coterminal Angles Answer: Sample answer: Example 3

Identify one positive and one negative angle coterminal with a 126o angle. B. 54°, –126° C. 234°, –54° D. 36°, –486° Example 3

Key Concept 4

Find Arc Length A. Find the length of the intercepted arc in a circle with a central angle measure of and a radius of 4 inches. Round to the nearest tenth. Arc Length r = 4 and Simplify. Example 4

The length of the intercepted arc is or about 4.2 inches. Find Arc Length The length of the intercepted arc is or about 4.2 inches. Answer: 4.2 in. Example 4

Find Arc Length B. Find the length of the intercepted arc in a circle with a central angle measure of 125° and a radius of 7 centimeters. Round to the nearest tenth. Method 1 Convert 125o to radian measure, and then use s = rθ to find the arc length. Example 4

Substitute r = 7 and . Arc length s = r r = 7 and Simplify. Find Arc Length Substitute r = 7 and . Arc length s = r r = 7 and Simplify. Example 4

Method 2 Use to find the arc length. Find Arc Length Method 2 Use to find the arc length. Arc length r = 7 and θ = 125° Simplify. The length of the intercepted arc is or about 15.3 centimeters. Answer: 15.3 cm Example 4

Find the length of the intercepted arc in a circle with radius 6 centimeters and a central angle with measure . A. 2.4 centimeters B. 4.7 centimeters C. 28.3 centimeters D. 45° Example 4

Key Concept 5

Find Angular and Linear Speeds A. RECORDS A typical vinyl record has a diameter of 30 cm. When played on a turn table, the record spins at revolutions per minute. Find the angular speed, in radians per minute, of a record as it plays. Round to the nearest tenth. Because each rotation measures 2π radians, revolutions correspond to an angle of rotation Example 5

Answer: 209.4 radians per minute Find Angular and Linear Speeds Angular speed Therefore, the angular speed of the record is or about 209.4 radians per minute. Answer: 209.4 radians per minute Example 5

A rotation of revolutions corresponds to an angle of rotation Find Angular and Linear Speeds B. RECORDS A typical vinyl record has a diameter of 30 cm. When played on a turn table, the record spins at revolutions per minute. Find the linear speed at the outer edge of the record as it spins, in centimeters per second. A rotation of revolutions corresponds to an angle of rotation Example 5

Linear Speed s = r minute Simplify. Find Angular and Linear Speeds Example 5

Find Angular and Linear Speeds Use dimensional analysis to convert this speed from centimeters per minute to centimeters per second. Therefore, the linear speed of the record is about 52.4 centimeters per second. Answer: about 52.4 cm/s Example 5

CAROUSEL Find the angular speed of a carousel in radians per minute if the diameter is 6 feet and it rotates at a rate of 10 revolutions per minute. A. 31.4 radians per minute B. 62.8 radians per minute C. 188.5 radians per minute D. 377.0 radians per minute Example 5

Key Concept 6

A. Find the area of the sector of the circle. Find Areas of Sectors A. Find the area of the sector of the circle. The measure of the sector’s central angle is , and the radius is 5 meters. Area of sector r = 5 and Example 6

Therefore, the area of the sector is or about 29.5 square meters. Find Areas of Sectors Therefore, the area of the sector is or about 29.5 square meters. Answer: Example 6

B. Find the area of the sector of the circle. Find Areas of Sectors B. Find the area of the sector of the circle. Convert the central angle measure to radians. Then use the radius of the sector to find the area. Example 6

Therefore, the area of the sector is or about 33.5 square feet. Find Areas of Sectors Area of sector r = 8 and Therefore, the area of the sector is or about 33.5 square feet. Answer: Example 6

Find the area of the sector of the circle. A. 7.9 in2 B. 15.7 in2 C. 58.9 in2 D. 117.8 in2 Example 6

LESSON 4–2 Degrees and Radians