1 / 16

13.3 – Radian Measures

13.3 – Radian Measures. 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 mi 4. diameter 3.4 ft 5. radius 5 mm 6. diameter 6.3 cm. Radian Measure.

clodia
Download Presentation

13.3 – Radian Measures

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 13.3 – Radian Measures

  2. 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 mi 4. diameter 3.4 ft 5. radius 5 mm 6. diameter 6.3 cm

  3. 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

  4. 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.

  5. 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.

  6. The Unit Circle

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

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

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

  10. 13 13 Find the degree measure of . 6 6 13 13 = Write a proportion. d° 180 6 6 radians • 180 = • dWrite the cross-product. d = Divide each side by . 30 13 • 180 6 • 1 An angle of radians measures 390°. Example = 390° Simplify.

  11. 90 . 180° – radians • = – radians • Multiply by radians 1 3 3 3 3 2 2 2 2 An angle of – radians measures –270°. 180° 180° radians radians Example Find the degree measure of an angle of – radians. = –270°

  12. 3 3 = radians Simplify. 3 5 4° • radians = 54° • radians Multiply by radians. 180° An angle of 54° measures radians. 10 180° 180° 10 10 Radian Measure Find the radian measure of an angle of 54°.

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

  14. 7 6 = 6 • Substitute 6 for r and for . 7 6 = 7 Simplify. 22.0 Use a calculator. 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.

  15. 1 4 Since one complete rotation (orbit) takes 4 h, the satellite completes of a rotation in 1 h. 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? Step 1: Find the radius of the satellite’s orbit. r = 6400 + 2900 Add the radius of Earth and the distance from Earth’s surface to the satellite. = 9300

  16. 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. 1 4 1 2 2 Step 3:  Find s for = . s = rUse the formula. = 9300 • Substitute 9300 for r and for . 14608 Simplify. 2 2 Radian Measure (continued) The satellite travels about 14,608 km in 1 h.

More Related