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19 problems
1
.
Use a graph to solve the equation on the interval
[−
2
π, 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) |
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 |
.
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
−1
< x ≤ ∞
−
π
≤ 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 x
sin−1
xsin x−1
| 1 | ||
|
csc x |
csc x
7
. Does arcsin x =
|
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 |
(b)
sin−1
−
2
2
10. Consider the function
y = arc
cos
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 |
|
− 0.2 |
0 | 0.2 | 0.4 |
| 0.6 | 0.8 |
(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 |
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
|
||||
|
cos−1
|
||||||
|
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 |
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)
