1. 6.4 Trigonometric Functions
Objectives:
Define the Trigonometric Ratios in the coordinate plane.
Define the Trigonometric functions in terms of the unit circle.

2. Example #1 Coterminal Angles
Coterminal angles share the same terminal side from standard position.
Find a positive & negative angle coterminal with the given angle.
A. 30° B. C.

3. Trigonometric Ratios on the Coordinate Plane
Regardless of the length of the radius, the trigonometric ratios remain the same.

4. The Unit Circle
Because the length of the radius doesn’t matter with respect to the trig ratios, the unit circle was developed to simplify calculations.
Therefore, any point P on the unit circle
has the coordinates (cos t, sin t).

5. Memorizing the Unit Circle

6. Reference Angles
Reference Angles are positive acute angles formed from the terminal side of θ and the x-axis.
They are used to simplify finding exact values for trigonometric ratios anywhere on the unit circle.

7. Example #2 Find the Following Reference Angles
-480°
290°
540° - 480° = 60°
360° - 290° = 70°

8. Example #2 Find the Following Reference Angles

9. Finding Exact Values for Trig Functions
(– , +)
(+, +)
Look at the sin(30°). The sine value is always the y-coordinate, so the exact value is ½. This angle is also a reference angle for 150°, 210°, 330°, etc.
If you look at the various values around the unit circle for the angles above, you’ll notice that they all have ½ as the value, but change signs depending on the quadrant.
Since (x, y)  (cosθ, sinθ), then the signs of the trig values will follow the signs of the x- and y-coordinates.
(– , –)
(+ , –)
Steps to finding exact values:
Find the reference angle.
Find the exact value for the trig function of the reference angle.
Translate the value to the correct quadrant changing the signs if necessary.

10. Example #3 Find the Exact Value for the Trig Functions
sin 120°
tan -405°
Reference:
Original angle is in Quadrant II.
The sine values stay positive in Quadrant II since they are the y-coordinates.
Reference:
Original angle is in Quadrant IV.
The tangent values become negative in Quadrant IV.

11. Example #3 Find the Exact Value for the Trig Functions
sec
csc
Reference:
Original angle is in Quadrant I.
Since cosine is positive, so is secant.
Reference:
Original angle is in Quadrant II.
Since sine is positive in Quadrant II, so is cosecant.