Math Analysis Problem Set 06-01 Write in the correct “good” unit circle angles. If no such unit circle angle exists, put a check mark if there’s some (ugly) angle for which it is true. Put an X if there is no angle for which it is true.

x

sin −1 ( x )

cos −1 ( x )

tan −1 ( x )

− 3 −1 − 3 2 − 2 2 − 3 3

−1 2 0

12 3 3 2 2 3 2

1

3

MA Notes 06 Problem Sets

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Math Analysis Problem Set 06-02 1. You should have three pictures in your mind when you think of sin −1 ( x ) :

a. Explain how each of the pictures relates to arcsin ( x ) .

⎛ 2⎞ b. Which do you find the most useful in mentally evaluating sin −1 ⎜ − ⎟? ⎝ 2 ⎠ c. Sketch the three pictures you should have in your mind when you think of cos −1 ( x ) and tan −1 ( x ) . 2. What is the difference between solving sin ( x ) = −1 and evaluating sin −1 ( −1) ? How does this relate to x 2 = 4 vs

4?

3. Use a graph to solve the equation 2sin ( 3x ) + 3 = 5 . 4. A Unit Circle is shown below. The grid lines are every 0.2 units.

a. Sketch an appropriate line on the circle and then estimate, in degrees, solutions to cos ( x ) = 0.4 . b. Sketch an appropriate line on the circle and then estimate, in degrees, solutions to sin ( x ) = −0.8 .

MA Notes 06 Problem Sets

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Math Analysis Problem Set 06-03 Are you incredibly fast at compositions? You should be once you figure out what’s going on here… 1. sin sin −1 ( −2 5 ) 2. sin sin −1 ( 6 7 ) 3. sin sin −1 ( −1 2 )

( ) 4. cos ( cos ( −1 5 )) 7. tan ( tan ( −2 3)) 10. csc ( sin ( −2 5 )) 13. sec ( cos ( −1 5 )) 16. cot ( tan ( −2 3)) 19. sin ( csc ( 5 4 )) 22. sin ( sin (π 6 )) 25. sin ( sin (11π 6 )) 28. cos ( cos ( 5π 4 )) 31. tan ( tan ( 2π 3)) 34. sin ( sin (π 9 )) 37. sin ( sin (17π 9 )) 40. cos ( cos (17π 12 )) 43. tan ( tan (15π 23)) −1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

( ) 5. cos ( cos ( 5 8 )) 8. tan ( tan (1 3)) 11. csc ( sin ( 6 7 )) 14. sec ( cos ( 5 8 )) 17. cot ( tan (1 3)) 20. cos ( sec ( 6 5 )) 23. sin ( sin ( 5π 6 )) 26. cos ( cos (π 4 )) 29. cos ( cos ( 7π 4 )) 32. tan ( tan ( 4π 3)) 35. sin ( sin ( 8π 9 )) 38. cos ( cos ( 5π 12 )) 41. cos ( cos (19π 12 )) 44. tan ( tan ( 31π 23)) −1

−1

−1

−1

−1

−1

−1

(

)

( ( )) tan ( tan ( −1 3 ))

6. cos cos −1 − 3 2 9.

−1

(

12. csc sin −1 ( −1 2 )

)

( ( )) 18. cot ( tan ( −1 3 )) 15. sec cos −1 − 3 2 −1

(

21. tan cot −1 ( 2 5 )

)

( ) ( cos ( 3π 4 )) ( tan (π 3)) ( tan ( 5π 3)) (sin (10π 9 )) ( cos ( 7π 12 )) ( tan ( 8π 23)) ( tan ( 38π 23))

24. sin −1 sin ( 7π 6 )

−1

27. cos −1

−1

30. tan −1

−1

33. tan −1

−1

36. sin −1

−1

39. cos −1

−1

42. tan −1

−1

45. tan −1

What do problems 1-9 illustrate?

What do problems 10-21 illustrate?

What do problems 22-30 illustrate?

What do problems 30-45 illustrate? [Hint: If you didn’t figure out anything for problems 22-30, I imagine these were very difficult for you…]

MA Notes 06 Problem Sets

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Math Analysis Problem Set 06-04 I love these types of problems… ⎛ ⎛ ⎛ ⎛ 3⎞ ⎞ ⎞ ⎞ 1. Evaluate sin ⎜ cos −1 ⎜ tan ⎜ sin −1 ⎜ ⎟ ⎟ ⎟ ⎟ . ⎝ 5⎠⎠⎠⎠ ⎝ ⎝ ⎝

⎛ ⎛ ⎛ ⎛ 5 ⎞⎞⎞⎞ 2. Evaluate tan ⎜ cos −1 ⎜ sin ⎜ cos −1 ⎜ − ⎟ ⎟ ⎟ ⎟ . ⎝ 13 ⎠ ⎠ ⎠ ⎠ ⎝ ⎝ ⎝ ⎛ ⎛ x⎞⎞ 3. Let f ( x ) = sin ⎜ cos −1 ⎜ ⎟ ⎟ . ⎝ 6⎠⎠ ⎝

a. State the domain of f ( x ) . b. Find an algebraic function for f ( x ) . c. Evaluate f ( 2 ) , f ( −3) , f ( −6 ) .

⎛ ⎛ 3x + 2 ⎞ ⎞ 4. Evaluate cot ⎜ sin −1 ⎜ . ⎝ 5 ⎟⎠ ⎟⎠ ⎝

MA Notes 06 Problem Sets

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Math Analysis Problem Set 06-05 1. Evaluate each of the following. ⎛ ⎛ ⎛ ⎛ 82π ⎞ ⎞ ⎞ ⎞ a. tan −1 ⎜ tan ⎜ cos −1 ⎜ sin ⎜ ⎝ ⎝ 5 ⎟⎠ ⎟⎠ ⎟⎠ ⎟⎠ ⎝ ⎝

⎛ ⎛ ⎛ ⎛ 92π ⎞ ⎞ ⎞ ⎞ b. cos −1 ⎜ cos ⎜ tan −1 ⎜ cot ⎜ ⎝ ⎝ 9 ⎟⎠ ⎟⎠ ⎟⎠ ⎟⎠ ⎝ ⎝

⎛ ⎛ ⎛ ⎛ ⎛ ⎛ 554π ⎞ ⎞ ⎞ ⎞ ⎞ ⎞ c. sin −1 ⎜ cos ⎜ cos −1 ⎜ sin ⎜ tan −1 ⎜ tan ⎜ ⎟ ⎝ ⎝ 13 ⎟⎠ ⎟⎠ ⎟⎠ ⎟⎠ ⎟⎠ ⎠ ⎝ ⎝ ⎝ ⎝

2. These two problems are essentially the same, solving them with graphs/thinking. Check them with a calculator. 1 a. Let f ( x ) = sin (π x ) and g ( x ) = x . For how many values of x, does 4 f ( x ) = g ( x ) ? Explain! 3 ⎛ πx⎞ b. Let f ( x ) = 3cos ⎜ ⎟ + 1 and g ( x ) = x + 6 . For how many values of x, does ⎝ 2 ⎠ 2 f ( x ) = g ( x ) ? Explain!

3. State the domain and range of each of the following functions. 5π ⎞ 4π ⎞ ⎛ 3π ⎛ x+ + 10 a. y = −8 csc ⎜ b. f ( x ) = −4 cot ⎜ 8x + ⎟ ⎟ +6 ⎝ 2 ⎝ 6 ⎠ 5 ⎠ 4. Establish the relationship between cos ( 6,835° ) and sin ( −7,827° ) . Clearly show each step along the way. 5. Without a calculator completely solve the right triangle MTV having right angle M, side t, with t = 18 , and V = 49° .

MA Notes 06 Problem Sets

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