1. Section 4.2 The Unit Circle

2. Has a radius of 1 Center at the origin Defined by the equations: a) b)

3. The Unit Circle The real number t corresponds to the distance around the unit circle. Each real number t corresponds to a point (x, y) on the unit circle. Sin t = Cos t = Tan t = (1,0)(-1,0) (0,1) (0,-1) t t x y 1 t y x

4. The Unit Circle Determine the exact values of the six trig functions of the angle Ѳ Ѳ

5. The Unit Circle Determine the exact values of the six trig functions of the angle Ѳ Ѳ

6. The Unit Circle

7. Find the point (x, y) on the unit circle that corresponds to the real number t. t = y = Sin = x = Cos = The number t corresponds to the point (, ) 1 2 30 º 60 º → Sin t = y, Cos t = x

8. The Unit Circle Find the point (x, y) on the unit circle that corresponds to the real number t. t = Sin t = Cos t = The real number t corresponds to the point

9. The Unit Circle Find the point (x, y) on the unit circle that corresponds to the real number t. Sin t = Cos t = Sin t = Cos t = t =

10. The Unit Circle

11. Section 4.2 The Unit Circle

12. Yesterday, we: a) Defined the unit circle b) Evaluated the exact value of the six trig functions of a point on the unit circle c) Found a point on the unit circle given the real number t d) Evaluated the sine, cosine, and tangent of the real number t Today: Evaluate all six trig functions of the real number t Use a functions period to evaluate the trig functions of t Use trig functions to evaluate other trig functions

13. The Unit Circle

15. Domain, Range, and Period Domain of the Sine function: All real numbers Domain of the Cosine function: All real numbers Range of these functions: [-1, 1] Period of both functions: 2π2π

16. Domain, Range, and Period What happens when you add 2 π to any value of t? Therefore: 1) Sin (t + 2 π ) = Sin t 2) Cos (t + 2 π ) = Cos t

17. Domain, Range, and Period Find the Sine Because = 2 π +, we have Sin = Sin (2 π + ) = Sin = ½

18. Domain, Range, and Period Find the Cosine Because = 2 π +, we have Cosin = Cos (2 π + ) = Cos = - ½

19. Practice CosSin = 2 π + → Cos = Cos → Cos = ½ Cos = ½ = 2 π + → Sin = Sin → Sin = Sin =

20. The Unit Circle

21. Even and Odd Functions Even FunctionsOdd Functions Cosine and Secant Cos (-t) = Cos t Sec (-t) = Sec t Sine and Cosecant Tangent and Cotangent Sin (-t) = - Sin t Csc (-t) = - Csc t Tan (-t) = - Tan t Cot (-t) = - Cot t

22. Even and Odd Functions Sin t =Cos t = - a) Sin (-t) = b) Csc (-t) = c) Sin ( π - t) = a) Sec (t) = b) Cos (t) = c) Cot (t + π ) = – – – –