1. Trigonometric Functions: The Unit Circle & Right Triangle Trigonometry
4.2/4.3
Trigonometric Functions:
The Unit Circle
&
Right Triangle Trigonometry

2. Quick Review

3. Quick Review Solutions

4. What you’ll learn about
How to identify a Unit Circle and its relationship to real numbers
How to evaluate Trigonometric Functions using the unit circle
Periodic Functions of sine and cosine functions
… and why
Extending trigonometric functions beyond triangle ratios opens up a new world of applications to model and solve real-life problems.

5. Initial Side, Terminal Side

6. Positive Angle, Negative Angle

7. Evaluating Trig Functions of a Nonquadrantal Angle θ
Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant.
Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ.
Draw a perpendicular segment from P to the x-axis, determining the reference triangle. If this triangle is one of the triangles whose ratios you know, label the sides accordingly. If it is not, then you will need to use your calculator.
Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant.
Use the coordinates of point P and the definitions to determine the six trig functions.

8. Trigonometric Functions of any Angle

9. Unit Circle
The unit circle is a circle of radius 1 centered at the origin.

10. Trigonometric Functions on Unit Circle

12. The 16-Point Unit Circle

26. Example Using one Trig Ratio to Find the Others

27. Example Using one Trig Ratio to Find the Others

44. Periodic Function