1. Find sec 5π/4

2. Use the Unit circle to prove Pythagorean identity
The Unit Circle and Trigonometry
Objective:
Use the unit circle to identify sine and cosine of an angle.
Use the Unit circle to prove Pythagorean identity

3. Plotting points on a coordinate plane
There are 4 quadrants
When plotting points:
We go horizontally first
Then Vertically
The points are (x,y)

4. The Unit Circle A unit circle is a circle with a radius of one
In trigonometry, the unit circle is centered at the origin.

5. Radius of one

6. The unit circle and right triangles
For the point (x,y) in Quadrant I, the lengths x and y become the legs of a right triangle whose hypotenuse is 1
By the Pythagorean Theorem, we have  x2 + y2 = 1. 

7. Example
X = 1 Y = 0
X = Y = .707
X = Y = .500

8. Angles on the unit circle
The radius of the circle always forms and angle with the x axis
This angle is referred to as
In the unit circle, we can use cosine and sine instead of x and y
Cos will always take place of X
Sin will always take place of Y

9. If we examine angle   in this unit circle, we can see that
Cosine is represented by the horizontal leg
Sine is represented by the vertical leg

10. Note  that       
becomes    

11. Quadrant 1

12. Examples What is cosine at 30° What is sine at 90° What is sine at 60°
What are sine and cosine at 0°

13. Calculator practice How do I type ½ into the calculator?
How do I type in the calculator
How do I type in the calculator?

14. Now lets use Pythagorean identity
Remember that becomes
Prove PI at 30°

15. Practice
Prove the Pythagorean identity at 0° 45°, 60°, and 90°

16. Lets try and solve for sine or cosine
At 30°, cosine is lets solve for sine.
At 90° sine is 1, lets solve for cosine

17. Practice
Using Pythagorean identity, solve for sine of 45° if cosine is
Using Pythagorean identity, solve for cosine of 60° if sine is
Using Pythagorean identity, solve for sine of 0° if cosine is 1

18. Common angles