13-3 Radian Measure

Quick Review

A central angle of a circle is an angle whose vertex is at the center of a circle and whose sides are radii of the circle. An intercepted arc is the portion of the circle whose endpoints are on the sides of the angle and whose remaining points lie in the interior of the angle. A radian is the measure of a central angle that intercepts an arc equal in length to a radius of the circle.

Example

What is the radian measure of an angle of − 210 ° ? negative 210 degrees question mark

− 210 ∘ = − 210 ∘ ⋅ π 180 ∘ radians = − 7 π 6 radians negative , 210 to the composition , equals negative , 210 to the composition , dot . fraction pi , over 180 to the composition end fraction . radians , equals negative , fraction 7 pi , over 6 end fraction . radians

Exercises

The measure θ of an angle in standard position is given.

1. Write each degree measure in radians and each radian measure in degrees rounded to the nearest degree.
2. Find the exact values of cos θ and sin θ for each angle measure.

13. 60°
14. − 45 ° negative 45 degrees
15. 180°
16. 2π radians
17. 5 π 6 radians fraction 5 pi , over 6 end fraction . radians
18. − 3 π 4 radians negative , fraction 3 pi , over 4 end fraction . radians

19. Use the circle to find the length of the indicated arc. Round your answer to the nearest tenth.

    

13-4 The Sine Function

Quick Review

The sine function y = sin θ y equals sine theta  matches the measure θ of an angle in standard position with the y-coordinate of a point on the unit circle. This point is where the terminal side of the angle intersects the unit circle. The graph of a sine function is called a sine curve.

For the sine function y = a sin b θ , y equals eh sine b theta comma  the amplitude equals | a | , vertical line eh vertical line comma  there are b cycles from 0 to 2π, and the period is 2 π b . fraction 2 pi , over b end fraction , .

Example

Determine the number of cycles the sine function y = − 7 sin 3 θ y equals negative 7 sine 3 theta  has in the interval from 0 to 2π. Find the amplitude and period of each function.

For y = − 7 sin 3 θ ,  a = − 7 y equals negative 7 sine 3 theta comma eh equals negative 7  and b = 3 . b equals 3 .  Therefore there are 3 cycles from 0 to 2π. The amplitude is | a | = | − 7 | = 7 . vertical line eh vertical line equals vertical line negative 7 vertical line equals 7 .  The period is 2 π b = 2 π 3 . fraction 2 pi , over b end fraction , equals , fraction 2 pi , over 3 end fraction . .

Exercises

Sketch the graph of each function in the interval from 0 to 2π.

20. y = 3 sin θ y equals 3 sine theta
21. y = sin 4 θ y equals sine 4 theta
22. Write an equation of a sine function with a > 0 , eh greater than 0 comma  amplitude 4, and period 0.5π

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