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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

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.

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.

The Unit Circle

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

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

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

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

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.

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

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

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

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