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Lesson Menu Five-Minute Check (over Lesson 4-1) 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

Over Lesson 4-1 5–Minute Check 1 Find the exact values of the six trigonometric functions of θ. A. B. C. D.

Over Lesson 4-1 5–Minute Check 2 If, find the exact values of the five remaining trigonometric functions of θ. A. B. C. D.

Over Lesson 4-1 5–Minute Check 3 A.a ≈ 26.8, c ≈ 13.8, C = 31 o B.a ≈ 19.7, c ≈ 11.8, C = 31 o C.a ≈ 11.8, c ≈ 19.7, C = 31 o D.a ≈ 15.1, c ≈ 17.3, C = 41 o Solve ΔABC. Round side lengths to the nearest tenth and angle measures to the nearest degree.

Over Lesson 4-1 5–Minute Check 4 Find the value of x. Round to the nearest tenth. A.37.1 B.32.5 C.15.7 D.8.7

Over Lesson 4-1 5–Minute Check 5 If, find tan θ. A. B. C. D.

Then/Now 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.

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

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

Example 1 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" = 35 o + 12'

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

Example 1 Write ° in DMS form. A.141°12'4.5" B.141.2°45'0" C.141°4'35" D.141°16'30"

Key Concept 2

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

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

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

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

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

Key Concept 3

Example 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 Find and Draw Coterminal Angles 80° + 360(–1)°= 80° – 360° or –280° Answer: 80 o + 360n o ; Sample answers: 440 o, –280 o

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

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

Key Concept 4

Example 4 Find Arc Length 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. Simplify. r = 4 and

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

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 1Convert 125 o to radian measure, and then use s = rθ to find the arc length.

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

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

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

Key Concept 5

Example 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 2 π radians, revolutions correspond to an angle of rotation

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

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

Example 5 Answer:about 52.4 cm/s 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.

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 radians per minute D radians per minute

Key Concept 6

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

Example 6 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 Convert the central angle measure to radians. Then use the radius of the sector to find the area.

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

Example 6 Find the area of the sector of the circle. A.7.9 in 2 B.15.7 in 2 C.58.9 in 2 D in 2

End of the Lesson