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Pre-Calculus Section 8.2: The Unit Circle and Its Functions (Trigonometry) 1. Trigonometric Functions: a: b: Let t be any real number and let P(x,y) be the terminal point on the unit circle determined by t. or 6. Example 1(E): Find the exact value of the trigonometric function at the given real number. or or 7. Unit Circle: 2. Example 1(A): Find the value of each a: sin(0) d: sin( ) b: cos( ) e: tan ( ) The unit circle is a circle with radius 1 centered at the origin in the xy plane. Its equation is defined by the following: c: tan( ) Note: Each (x,y) coordinate can also be defined by (cos(t), sin(t)), where t = terminal point as determined by the arc of the unit circle 3. Example 1(B): Find the value of each a: sin( ) d: cos( ) b: cos( ) e: sin( ) Example: t = produces the point and c: tan( ) 4. Example 1(C): Find the value of all six trig functions for each real number t given below: a: Additionally: Due to the sign values of (x,y) in the coordinate plane, the value of (cos(t), sin(t)) will maintain the same sign as (x,y). Some people remember sign values by using the saying, “All, Scholars, Take, Calculus” b: 5. Example 1(D): Find the value of all six trig functions for each real number t given below: 8. Example 2(A): Find the sign of the expression if the terminal point determined by t is in the given quadrant. Quadrant I A: sin(t) B: cos(t) Quadrant II C: tan(t) D: sec(t) 12. Example 3 (C): Find the exact values of the Quadrant III E: csc(t) trigonometric functions and . F: cot(t) Quadrant IV: G: sin(t) 13. Example 3 (D): Find the exact values of the H: sec(t) trigonometric functions 9. Example 2 (B): From the information given, find the quadrant in which the terminal point determined by t lies. and . 14. Example 3 (E): Find the exact values of the trigonometric functions and . and 15. Example 3 (F): Find the exact values of the 10. Example 3 (A): Find the exact values of the trigonometric functions trigonometric functions and and 16. Example 3 (G): Find the exact values of the a. trigonometric functions and b. c. 17. Example 3 (H): Find the exact values of the trigonometric functions 11. Example 3 (B): Find the exact values of the trigonometric functions and and . 18. Example 3 (I): Find the exact values of the trigonometric functions and 19. Example 3 (J): Find the exact values of the trigonometric functions and . (a) By using the figure, estimate (b) By using a calculator in radians find 20. Example 3(K): Find the value of each a: sin( ) b: cos( c: tan( d: cos( ) e: sin( ) ) 23. Example 4 (C): Find the approximate value of ) 21. Example 4 (A): Find the approximate value of (a) By using the figure. Please give the answer to one decimal place. __________ (a) By using the figure find: (b) By using a calculator in radians find: 22. Example 4 (B): Find the approximate value of (b) By using a calculator in radians. Please give the answer to five decimal places. __________ 24. Even-Odd Properties: Sine, Cosecant, Tangent, and Cotangent are Odd Functions: sin(-t) = -sin(t) csc(-t) = -csc(t) 25. Example 5 (A): Determine whether the function tan(-t) = -tan(t) cot(-t) = -cot(t) is even, odd, or neither. Cosine and Secant are Even Functions: a. neither b. odd c. even Example 5 (B): Determine whether each function is even, odd, or neither. Match each function with the corresponding concept. cos(-t) = cos(t) a. sec(-t) = sec(t) c. b. 26. even 28. neither 27. odd 29. Example 6 (A): Find the values of the trigonometric functions of t if and the terminal point of t is in quadrant II. 30. Example 6 (B): Find the values of the trigonometric functions of t if and the terminal point of t is in quadrant III. 31. Example 6 (C): Find the values of the trigonometric functions of t if 32. Example 6 (E): Find the values of the trigonometric functions of t if in quadrant IV. 33. Pythagorean Identities: and . and the terminal point of t is 34. Example 7(A): Write sin(t) in terms of cos(t), where t is in Quadrant II. 35. Example 7(B): Write cos(t) in terms of sin(t), where t is acute. 36. Example 7(C): Write tan(t) in terms of cos(t), where t is in Quadrant III. 37. Example 7(D): Write sec( ) in terms of cos( ), where is in Quadrant IV. 38. Example 7(E): Write sec( ) in terms of sin( ), where is in Quadrant IV. 39. Example 7(F): Write tan( ) in terms of sin( ), where is in Quadrant III. 40. Example 7(G): Write sin( ) in terms of cot( ), where is in Quadrant IV.