13-1 Exploring Periodic Data

Quick Review

A periodic function repeats a pattern of y-values at regular intervals. One complete pattern is called a cycle. A cycle may begin at any point on the graph. The period of a function is the length of one cycle. The amplitude of a periodic function is half the difference between its maximum and minimum values.

Example

What is the period of the periodic function?

One cycle is 5 units long, so the period of the function is 5.

Exercises

6. Determine whether the function below is or is not periodic. If it is, identify one cycle in two different ways and find the period and amplitude.

   

7. Sketch the graph of a wave with a period of 2 and an amplitude of 4.
8. Sketch the graph of a wave with a period of 4 and an amplitude of 3.

13-2 Angles and the Unit Circle

Quick Review

An angle is in standard position if the vertex is at the origin and one ray, the initial side, is on the positive x-axis. The other ray is the terminal side of the angle. Two angles in standard position are coterminal if they have the same terminal side.

The unit circle has radius of 1 unit and its center at the origin. The cosine of θ ( cos  θ ) open , cos , theta close is the x-coordinate of the point where the terminal side of the angle intersects the unit circle. The sine of θ ( sin  θ ) open , sin , theta close is the y-coordinate.

Example

What are the cosine and sine of − 210 ° ? negative 210 degrees question mark

Sketch an angle of − 210 ° negative 210 degrees  in standard position with a unit circle. The terminal side forms a 30° − 60 ° − 90 ° negative 60 degrees negative 90 degrees  triangle with hypotenuse = 1, shorter leg = 1 2 , longer leg = 3 2 1 half , comma . longerleg . equals , fraction square root of 3 , over 2 end fraction  Since the terminal side lies in Quadrant II, cos ( − 210 ° ) cosine open negative 210 degrees close  is negative and sin ( − 210 ° ) sine open negative 210 degrees close  is positive.

cos ( − 210 ∘ ) = − 3 2 and  sin ( − 210 ∘ ) = 1 2 cosine . open . negative , 210 to the composition . close . equals negative , fraction square root of 3 , over 2 end fraction , and sine . open . negative , 210 to the composition . close . equals , 1 half

Exercises

9. Find the measurement of the angle in standard position below.

   

10. Sketch a − 30 ° negative 30 degrees  angle in standard position.
11. Find the measure of an angle between 0° and 360° coterminal with a − 120 ° negative 120 degrees  angle.
12. Find the exact values of the sine and cosine of 315° and − 315 ° . negative 315 degrees .  Then find the decimal equivalents. Round your answers to the nearest hundredth.

Table of Contents