Angle Measures in Degrees & Radians Trigonometry 1.0 Students understand the notation of angle and how to measure it, in both degrees and radians. They.

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Angle Measures in Degrees & Radians Trigonometry 1.0 Students understand the notation of angle and how to measure it, in both degrees and radians. They can convert between degrees and radians.  O A B

- (the side where the  begins) – is always the positive x-axis.  O A B The initial side (OA) The vertex is always at the origin. The terminal side (OB) - is the ray that forms the  An angle is in standard position when: 1. The initial side is the positive x-axis 2. The vertex is at the origin. Angles in Standard Position

x y initial side terminal side vertex Measuring Angles

Degrees? 180˚ The terminal side ends up in quadrant __. Positive  s are drawn counterclockwise. Draw a 135˚ . 90˚ 270˚ 0˚ Start on the positive x-axis. Quadrants? III IIIIV or 360˚ 135˚ II

Negative angles are drawn clockwise. (Start on the positive x-axis.) - 60˚ What Quadrant? ___IV - 210˚ What Quadrant? __II 0˚ -90˚ -180˚ -270˚ or -360˚ -60˚ -210˚

Radian Measure The distance around a circle is 360°. x y r The distance around a circle is also 2πr. So, 2πr = 360°. In trigonometry, we deal with a “unit circle” where the radius is 1. Therefore: 2π = 360° or π = 180° That’s radian measure!

Unit Circle x y

30° 300 ° 80° 810° To change radians to degrees, multiply by. You try it:

To change degrees to radians, multiply by. 60˚ = ___ 20˚ = __ 80˚ = ___45˚ = __ You try it:

Coterminal Angles in Radians Angle has measure of 9π/4 (405°) Angle has measure of -7π/4 (-315°) Angle has measure of π/4 (45°) To find coterminal angles in radians, add or subtract 2π. Coterminal Angles have the same initial side the same vertex the same terminal side but different measures

Find two coterminal angles, one positive and one negative. 2π/3 - 5π/7 15π/4 PositiveNegative 8π/3 9π/7 7π/4 -4π/3 -19π/7 -π/4 ± 6π/3 ± 14π/7 - 8π/4

Find two coterminal angles, one positive and one negative for 140°. To find coterminal angles in degrees: Add 360° or Subtract 360° 140° 140° + 360° = 500° 140° - 360° = -220° y

Find two coterminal angles, one positive and one negative. 320° - 245° 880° PositiveNegative 680 ° -40 ° 115 ° 160 ° -605 ° -200 ° ± 360° - 720°- 360°

Complementary & Supplementary Angles Complementary angles add to 90° or Supplementary angles add to 180° or

If possible, find the complement and supplement of the angle. 70° ComplementSupplement 20 ° 110 ° 90°- 70° 180°- 70° none

Arc Length s = rθ arc length = radius · angle (in radians) s r θ Determine the arc length of a circle of radius 6 cm intercepted by an angle of π/2. s = (π/2)·6 s = 3π cm

If the central angle is given in degrees, change it to radians in the problem! Find the arc length to the nearest tenth of a centimeter of a circle of radius 7 cm that is intercepted by a central angle of 85°. s = 7(85)(π/180) s = 10.4 cm

Homework Page Memorize the unit circle!