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Warm-up

Warm-up. 1. Sketch a 125° angle. 2. Find the coterminal angle between 0° and 360° for 600°. 3. Using the unit circle find the following: a. sin 210° b. cos 315° . Section 13.3. Radian Measure. 1 st Day. A central angle of a circle is an angle with a vertex at the center of a circle.

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Warm-up

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  1. Warm-up • 1. Sketch a 125° angle. • 2. Find the coterminal angle between 0° and 360° for 600°. • 3. Using the unit circle find the following: • a. sin 210° • b. cos 315°

  2. Section 13.3 Radian Measure

  3. 1st Day

  4. A central angle of a circle is an angle with a vertex at the center of a circle. • An intercepted arc is the portion of the circle with endpoints on the sides of the central angle and remaining points within the interior of the angle. intercepted arc central angle

  5. When a central angle intercepts an arc that has the same length as a radius of the circle, the measure of the angle is defined to be one radian. Like degrees, radians measure the amount of rotation from the initial side to the terminal side of an angle. r 1 radian r

  6. Because the circumference of acircle is 2πr, there are 2π radians in any circle. • Since 2π radians = 360°, therefore • π radians = 180°, you can use a proportion such as • to convert between degrees and radians.

  7. Example • a. Find the radian measure of an angle of 60°. • 1. Write a proportion. • 2. Write the cross products.

  8. 3. Divide each side by 180. • 4. Simplify to the exact answer and then find the answer rounded to two decimal places.

  9. Converting Between Radians and Degrees

  10. Example

  11. End of the 1st Day

  12. 2nd Day

  13. You can find the sine and cosine of angles in radian measures by first converting the radian measure to degrees and then using the unit circle.

  14. Example

  15. 1. Convert from radians to degrees. • 2. Look on the unit circle for 45°. The x-coordinate is the value of the cosine and the y-coordinate is the value of the sine.

  16. You can find the length of an intercepted arc by using the proportion • When you simplify this proportion we get the formula for finding the length of an intercepted arc.

  17. Length of an Intercepted Arc • For a circle of radius r and a central angle of measure θ (in radians), the length s of the intercepted arc is s = rθ. s r θ r

  18. Example • Use the circle at the right. Find length s as an • exact answer and then • rounded to the nearest • tenth. s 3 inches

  19. 1. Use the formula s = rθ. s 3 inches

  20. Example • A weather satellite in a circular orbit around Earth completes one orbit every 2 hours. The radius of Earth is about 6400 km, and the satellite orbits 2600 km above Earth’s surface. How far does the satellite travel in 1 hour?

  21. 1. Since 1 revolution • takes 2 hours, the • satellite completes • in 1 hour. • 2. Find the radius of the satellite orbit. • r = 6400 + 2600 = 9000 km. Earth 6400 km

  22. 3. Find the measure of the central angle the satellite travels through 1 hour. • 4. Find s as an exact answer and then round to the nearest kilometer. • s = rθ • s = 9000π km. • s ≈ 28,274 km.

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