6.4 Trigonometric Functions

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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.