1 / 24

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

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 33 0 angle with the ground, how high is the kite?. h. h. h. Angle of elevation. 40’. 40’. 33 0.

orde
Télécharger la présentation

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

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. 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

More Related