1. Trigonometric Functions: The Unit Circle
Skill 26

2. Objectives…
Identify a unit circle and describe its relationship to real numbers.
Evaluate trigonometric functions using the unit circle.
Use domain and period to evaluate sine and cosine functions and use a calculator to evaluate trigonometric functions.

3. The unit circle is given by the equation
x2 + y2 = 1
It helps to solve trigonometric functions.

4. The Unit Circle
Imagine that the real number line is wrapped around this circle, with positive numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping, as shown.

5. The Unit Circle
In general, each real number t also corresponds to a central angle  (in standard position) whose radian measure is t. With this interpretation of t, the arc length formula s = r (with r = 1) indicates that the real number t is the (directional) length of the arc intercepted by the angle  given in radians.

6. The Trigonometric Functions
The coordinates x and y are two functions of the real variable t. Use these coordinates to define the six trigonometric functions of t.
sine (sin)
cosine (cos)
tangent (tan)
cosecant (csc)
secant (sec)
cotangent (cot)

7. Need to Know… - Exact Answers (no decimals)
- Points are in relation to their given angle
- Each point: (cos t , sin t)

8. The Trigonometric Functions

10. Example– Evaluate the 6 trig functions

11. Example– Evaluate the 6 trig functions

12. Example– Evaluate the 6 trig functions

13. Example– Evaluate the 6 trig functions

14. Domain and Period of Sine and Cosine
The domain of the sine and cosine functions is the set of all real numbers.
To determine the range of these two functions, consider the unit circle

15. Domain and Period of Sine and Cosine
The values of sin(t + 2) and cos(t + 2) correspond to those of sin t and cos t.
sin(t + 2 n) = sint and cos(t + 2 n) = cost
for any integer n and real number t. Functions that behave in such a repetitive (or cyclic) manner are called periodic.

16. Domain and Period of Sine and Cosine

17. Example–Using the Period to Evaluate Sine and Cosineb.

18. 26: Trig. Functions and The Unit Circle
Summarize Notes
Videos
Homework
Worksheet
Quiz