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13.3 – Radian Measures

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Radian Measure Find the circumference of a circle with the given radius or diameter. Round your answer to the nearest tenth. 1.radius 4 in.2.diameter 70 m 3.radius 8 mi4.diameter 3.4 ft 5.radius 5 mm6.diameter 6.3 cm

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Radian Measure 1. C = 2 r = 2 (4 in.) 25.1 in. 2.C = d = (70 m) 219.9 m 3.C = 2 r = 2 (8 mi) 50.3 mi 4.C = d = (3.4 ft) 10.7 ft 5.C = 2 r = 2 (5 mm) 31.4 mm 6.C = d = (6.3 cm) 19.8 cm Solutions

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Vocabulary and Definitions A central angle of a circle is an angle with a vertex at the center of the circle. An intercepted arc is the arc that is “captured” by the central angle.

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Vocabulary and Definitions When the central angle intercepts an arc that has the same length as a radius of the circle, the measure of the angle is defined as a radian. r r Like degrees, radians measure the amount of rotation from the initial side to the terminal side of the angle.

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The Unit Circle

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The “Magic” Proportion This proportion can be used to convert to and from Degrees to Radians. Degrees° 180° = r radians radians Example: Find the radian measure of angle of 45°. Write a proportion. 45° 180° = r radians radians An angle of 45° measures about 0.785 radians. Write the cross-products. 45 = 180 r Divide each side by 45.r = 45 180 = 0.785Simplify. 4

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The “Magic” Proportion This proportion can be used to convert to and from Degrees to Radians. Degrees° 180° = r radians radians Example: Find the radian measure of angle of -270°. Write a proportion. -270° 180° = r radians radians An angle of -270° measures about -4.71 radians. Write the cross-products. -270 = 180 r Divide each side by 45.r = -270 180 -4.71Simplify. 2 -3

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Let’s Try Some Convert the following to radians a.390 o b. 54 o c. 180 o

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Example = 390°Simplify. Find the degree measure of. 6 13 Write a proportion. 6 13 radians = d° 180 180 = dWrite the cross-product. 6 13 d = Divide each side by. 13 180 6 1 30 An angle of radians measures 390°. 6 13

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Example Find the degree measure of an angle of – radians. 2 3 = –270° An angle of – radians measures –270°. 2 3 – radians = – radians 2 3 180° radians 2 3 180° radians 1 90 Multiply by 180° radians.

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Radian Measure Find the radian measure of an angle of 54°. 5 4° radians = 54° radians Multiply by radians. 180° 3 10 3 radians=Simplify. An angle of 54° measures radians. 10 3

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Draw the angle. Radian Measure Find the exact values of cos and sin. radians 3 3 radians = 60° Convert to degrees. 3 180° radians Complete a 30°-60°-90° triangle. The hypotenuse has length 1. radians 3 Thus, cos = 1212 and sin radians 3 =. 3 2 The shorter leg is the length of the hypotenuse, and the longer leg is 3 times the length of the shorter leg. 1212

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Radian Measure Use this circle to find length s to the nearest tenth. s = r Use the formula. The arc has length 22.0 in. = 7Simplify. 22.0Use a calculator. = 6 Substitute 6 for r and for . 7 6 7 6

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Radian Measure Another satellite completes one orbit around Earth every 4 h. The satellite orbits 2900 km above Earth’s surface. How far does the satellite travel in 1 h? Since one complete rotation (orbit) takes 4 h, the satellite completes of a rotation in 1 h. 1414 Step 1: Find the radius of the satellite’s orbit. r = 6400 + 2900Add the radius of Earth and the distance from Earth’s surface to the satellite. = 9300

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Radian Measure (continued) The satellite travels about 14,608 km in 1 h. Step 2: Find the measure of the central angle the satellite travels through in 1 h. = 2 Multiply the fraction of the rotation by the number of radians in one complete rotation. = Simplify. 1414 1212 Step 3: Find s for =. s = r Use the formula. = 9300 Substitute 9300 for r and for. 14608Simplify. 2 22

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