precalculus

1. Use a graph to solve the equation on the interval [−2π, 2π] .(List the solutions in increasing order from left to right on the x-axis.)

csc(x) = √2

x =  (smallest value)

x =

x =

x =  (largest value)

2. Use a graph to solve the equation on the interval [−2π, 2π]. (List the solutions in increasing order from left to right on the x-axis.)

tan x = 1

x =  (smallest value)

x =

x =

x =  (largest value)

3. Use a graph to solve the equation on the interval [−2π, 2π].

tan x = 

4. To get more solutions of 

tan x = 

3

, add π or 

−π

 successively to 

x = 

π

3

.

  Therefore, the solutions of the equation 

tan x = 

3

 in the interval 

[−2π, 2π],

 are as follows. (List the solutions in increasing order from left to right on the x-axis.)

x =  (smallest value)

x =

x =

x =  (largest value)

5. Fill in the blanks.

Function	Alternative Notation	Domain	Range
y = arcsin x	y =	−1 < x ≤ ∞	−  π 2  ≤ y ≤  π 2
		−1 ≤ x ≤ 1
		−∞ < x ≤ 1
		−∞ < x < ∞
		−1 < x < 1

6. What notation can you use to represent the inverse sine function? (Select all that apply.)

arcsin xsin−1 xsin x−1

1

csc x

csc x

7. Does arcsin x = 

1

sin x

?

YesNo    

8. Find the exact value of each expression without using a calculator.

(a)    

arctan 1

  (b)    

arctan 0

9. Find the exact value of each expression without using a calculator.

(a)    

cos−1 

− 

2

2

  (b)    

sin−1 

− 

2

2

 

 

10. Consider the function 

y = arccos x.

(a) Use a graphing utility to complete the table. (Round your answers to four decimal places.)

x	−1	−0.8	−0.6	−0.4
y

x	−0.2	0	0.2	0.4
y

x	0.6	0.8	1
y

(b) Plot the points from the table in part (a) and graph the function. (Do not use a graphing utility.)

(c) Use the graphing utility to graph the inverse cosine function and compare the result with your hand-drawn graph in part (b).

(d) Determine any intercepts of the graph.

x-intercept	(x, y) =
y-intercept	(x, y) =

What type of symmetry does the graph have?

originx-axis    noney-axis

11. Use a calculator to approximate the value of the expression. Round your answer to the nearest hundredth. (Enter your answer in radians.)

cos−1 0.79

12. Use a calculator to approximate the value of the expression. Round your answer to the nearest hundredth. (Enter your answer in radians.)

arccos(−0.7)

13. Use a calculator to approximate the value of the expression. Round your answer to the nearest hundredth.

tan-1 (6.3)

14. Use an inverse trigonometric function to write θ as a function of x. θ =  

15. Use an inverse trigonometric function to write θ as a function of x.  θ =  

16. Consider the following.

Find the length of the third side of the triangle in terms of x.   Find θ in terms of x for all three inverse trigonometric functions.

θ	=	sin−1
θ	=	cos−1
θ	=	tan−1

17. Use the properties of inverse functions to find the exact value of the expression. (Enter your answer in radians.)

cos−1

cos 

3π

2

 

18. Find the exact value of the expression. (Hint: Sketch a right triangle. Enter your answer in radians.)

tan

arcsin

− 

6

7

 

 

19.  Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 7.)

sin(arctan x)