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Aim: What are radians and how do they differ from degrees?

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  1. Aim: What are radians and how do they differ from degrees? Do Now: A kite is held by a taut string pegged to the ground. If the string is 40 feet long and makes a 330 angle with the ground, how high is the kite?

  2. h h h Angle of elevation 40’ 40’ 330 A kite is held by a taut string pegged to the ground. If the string is 40 feet long and makes a 330 angle with the ground, how high is the kite? Sin 330 = .544639. . . = 40 feet h 21.8 feet The kite is approximately 22 feet off the ground.

  3. Circle - Central Angles B 900 A O 1800 Central Angle – of a circle is an angle whose vertex is the center of the circle. The sum of all central angles of a circle is 3600.

  4. Given: radius r Arc AB = r 1 radian If the length of arc AB measures r, then the measure of central angle BOA is 1 radian. 1 2 3 4 5 6 .28 Radians Definition: A radian is the measure of the central angle that intercepts an arc equal in length to the radius of the circle. B r A r O • radians = 1800 2  radians = 3600 r

  5. 2 2 units 2 2 1 2 1 2 2 Radians - Unit Circle If the measure of arc AB is 1 unit, then the measure of central angle BOA is 1 radian. Given: radius = 1 Arc AB = 1 B 1 1 radian A 1 O

  6. length of intercepted arc length of radius Finding the Measure of Central Angles in Radians B s  A r O measure of angle  in radians =

  7. s • Arc AB defines • a semi-circle • with • length s. • is the central angle whose measure is 1800.  B A r O s = 1/2 C = Radians & Degrees - Semicircle C = pD = p2r 1/2 (2πr) = πr = 1800

  8. radians = 900 length of intercepted arc s radians = 2700 length of radius r Measure of Central Angles in Radians measure of angle  in radians = B s  A O r π radians = 1800 2π radians = 3600

  9. π π Degrees in 1 Radian B r 1 radian  A r O π radians = 1800 or 57º18’ (to the nearest minute)

  10. What is the complement of  = /12? radians = 900 Supplementary, Complementary & Coterminal Angles π radians = 1800 2π radians = 3600 What is the supplement of  = 5/6? Name, in terms of radians, a coterminal angle for  = 17/6

  11. r = 4 mCentral = 1.5 radians substitute the given values Model Problem In a circle, the length of a radius is 4 centimeters. Find the length of an arc intercepted by a central angle whose measure is 1.5 radians. solve for s s = 6 cm.

  12. Model Problem A weather satellite in a circular orbit around Earth completes one orbit every 3 hours. The radius of the Earth is about 6400 km, and the satellite is positioned 2600 km above Earth. How far does the satellite travel in 1 hour. s since one complete rotation takes 3 hr., the satellite completes 1/3 of a rotation in 1hr. r s = distance traveled in 1 hr. r = 6400 + 2600 = 9000

  13. Proportion: x = degrees solve for x: Changing from Radians to Degrees How do we convert a π/4 radians to degrees? Method 1 πx = 45π x = 45 degrees

  14. Substitution: π radians = 1800 Changing from Radians to Degrees How do we convert a π/4 radians to degrees? Method 2

  15. mA in degrees measure of a straight angle in degrees = mA in radians measure of a straight angle in radians Proportion: identify variable: x = measure in radians of 750 & set up proportion solve for x Degrees to Radians – Model Problem Convert a 750 to radians? Use the fact: π radians = 1800 180x = 75π

  16. Convert radians to degrees. Method 1 Proportion: x = degrees in angle of 7π/3 radians solve for x: Model Problem πx = 420π x = 420 degrees

  17. Convert radians to degrees. Model Problem (con’t) Method 2 Substitution: π radians = 1800

  18. π radians = 1800 Proportion: identify variable: x = measure in radians of 1350 & set up proportion solve for x Model Problems Convert 1350 to radians? 180x = 135π

  19. Regents Prep What is the number of degrees in an angle whose radian measure is • 150 2. 165 • 3. 330 4. 518 Find to the nearest minute, the angle whose measure is 3.45 radians. x = 197.6704o x = 197o 40’ convert .6704o to minutes m’ = 40.2’

  20. Model Problem In a circle, a central angle of 1/3 radian intercepts an arc of 3 centimeters. Find the length, in centimeters, of a radius of the circle. If  is 4 and r = 1.25, find s.

  21. Model Problem If f(x) = cos 2x and sin x, find f(/2) substitute simplify evaluate If f(x) = sinx cos2x, find f(/3)

  22. convert to radians Finding Arc Length A circle has a radius of 4 inches. Find the length of an arc cut off by a central angle of 2400. substitute r = 4 into s = r

  23. Regents Prep A circle has a radius of 4 inches. In inches, what is the length of the arc intercepted by a central angle of 2 radians? 1. 2 2. 2 3. 8 4. 8