13.6 Circular Functions Objectives: Define and use the trig functions based on the unit circle. Find the exact values of trig functions of angles.

Unit Circle We are going to define trig values in terms of the unit circle. All of the values from the chart repeat throughout the other 3 quadrants. Remember that sin θ=y/r, cos θ=x/r, tan θ=y/x, csc θ=r/y, sec θ=r/x, and cot θ=x/y. On the unit circle, r=1, so sin θ=y and cos θ=x. Use this fact and the chart of values to write ordered pairs for the unit circle.

Definition of Sine and Cosine If the terminal side of an angle θ in standard position intersects the unit circle at P(x,y), then cos θ=x and sin θ=y. Therefore, the coordinates of P can be written as P(cos θ, sin θ) P(cos θ, sin θ) (0,1) (1,0) (0,-1) (-1,0)

Example Given point P on the unit circle, find sin θ and cos θ . 1. 2.

The Unit Circle with Coordinates Remember that sin θ=y and cos θ=x. An angle and its reference angle have the same coordinate – just different signs. For example 30° and 150° have the same reference angle and their coordinates are and

Graph of Sine If the same information in the unit circle is presented on the coordinate plane where the horizontal axis shows the value of θ and the vertical axis shows the value of sin θ, the graph would look like: 90° 180° 270° 360° -90° -180° -270° -360° 1 -1

Graph of Cosine For the cosine graph: -360 -270 -180 -90 90 180 270

Periodic Function Periodic functions are called such because they repeat a pattern in a specific distance on the graph. Remember that the graphs come from the unit circle, and that the values repeat themselves after each rotation. The period is the distance from the beginning to the end (when it starts to repeat itself) The parent graph of sin θ and cos θ has a period of 360° or 2π because for each 360°, the values are the same.

Finding values using the unit circle Take out each revolution (subtract 360 or 2π) Look at unit circle for the coordinate at that location. Remember sinθ=y, cosθ=x. Ex: cos 690° 690-360=330 cos 690°=

Find the exact value of each function. Sin 570° 5(sin 45°)(cos 45°)

Determine the period of each function 1. 2.

Homework p 743 12-32 even