1. Math IV Warm Up Draw a circle on your paper. Fill in the degrees of the entire unit circle

8. Math IV Lesson 22 Essential Questions: How do you evaluate trigonometric functions using the unit circle? What is the unit circle? Standards: MM4A2. Students will use the circle to define the trigonometric functions. a. Define and understand angles measured in degrees and radians, including but not limited to 0°, 30°, 45°, 60°, 90°, their multiples, and equivalences b. Understand and apply the six trigonometric functions as functions of general angles in standard position. c. Find values of trigonometric functions using points on the terminal sides of angles in the standard position. d. Understand and apply the six trigonometric functions as functions of arc length on the unit circle. e. Find values of trigonometric functions using the unit circle.

9. Vocabulary A circle with center at (0, 0) and radius 1 is called a unit circle.

10. The equation of this circle would be So points on this circle must satisfy this equation. (1,0) (0,1) (0,-1) (-1,0)

11.  Equation of the unit circle X 2 + y 2 = 1 Degrees/ radians quiz tomorrow

12. A Unit Circle has Radians, degrees and coordinates

13. Let's pick a point on the circle. We'll choose a point where the x is 1/2. If the x is 1/2, what is the y value? (1,0) (0,1) (0,-1) (-1,0) x = 1/2 You can see there are two y values. They can be found by putting 1/2 into the equation for x and solving for y. We'll look at a larger version of this and make a right triangle.

14. Every point on the unit circle has a x value and a y value. Let t be any angle in radians on the unit circle

15. The Trigonometric Functions The coordinates x and y are two functions of the real variable t. You can use these coordinates to define the six trigonometric functions of t. sine cosine tangent cosecant secant cotangent These six functions are normally abbreviated sin, cos, tan, csc, sec, and cot, respectively.

16. The Trigonometric Functions Let t be a real number and let (x,y) be the point on the unit circle corresponding to t Sin(t) = y csc(t) = 1/y Cos(t) = x sec(t) = 1/x Tan(t) = y/xcot(t) = x/y

17. Example 1 – Evaluating Trigonometric Functions Evaluate the six trigonometric functions at each real number. a. b. c. d. Solution: For each t-value, begin by finding the corresponding point (x, y) on the unit circle. Then use the definitions of trigonometric functions.

18. Example1(a) – Solution t =  /6 corresponds to the point cont’d

19. Example1(b) – Solution t = 5 /4 corresponds to the point cont’d

20. Example1(c) – Solution t =  corresponds to the point (x, y) = (– 1, 0). cont’d

21. Example1(d) – Solution Moving clockwise around the unit circle, it follows that t = – /3 corresponds to the point cont’d

22. Heroic Quest convert all of the degrees to radians

23. Evaluating trigonometric functions using the unit circle