Find sec 5π/4

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

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

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

Radius of one

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. 

Example X = 1 Y = 0 X = .707 Y = .707 X = .866 Y = .500

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

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

Note  that        becomes    

Quadrant 1

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

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

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

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

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

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

Common angles