Find values of the inverse trigonometric functions.
Simplify expressions such as (sin-1x)
Simplify expressions involving compositions such as (cos-112) without using a calculator.
Simplify expressions such as (a/b) by making a drawing and reading off appropriate ratios.
The graphs of the sine, cosine, and tangent functions are shown on the following page. Are the inverses of these functions also functions? We learned earlier that a function has an inverse that is a function if it is one-to-one, which we can check with the horizontal-line test.
Note that for each function, a horizontal line (shown in red) crosses the graph more than once. Therefore, none of them has an inverse that is a function.
The graphs of an equation and its inverse are reflections of each other across the line y=x. Let’s examine the graphs of the inverses of each of the three functions graphed above.
We can check again to see whether these are graphs of functions by using the vertical-line test. In each case, there is a vertical line (shown in red) that crosses the graph more than once, so each fails to be a function.
Recall that a function like f(x)=x2 does not have an inverse that is a function, but by restricting the domain of f to nonnegative numbers, we have a new squaring function, f(x)=x2,x≥0, that has an inverse, f-1(x)=√x. This is equivalent to restricting the range of the inverse relation to exclude ordered pairs that contain negative numbers.
In a similar manner, we can define new trigonometric functions whose inverses are functions. We can do this by restricting either the domains of the basic trigonometric functions or the ranges of their inverse relations. This can be done in many ways, but the restrictions illustrated below with solid red curves are fairly standard in mathematics.
For the inverse sine function, we choose a range close to the origin that allows all inputs on the interval [-1,1] to have function values. Thus we choose the interval [-π/2, π/2] for the range (Fig. 1). For the inverse cosine function, we choose a range close to the origin that allows all inputs on the interval [-1,1] to have function values. We choose the interval [0, π] (Fig. 2). For the inverse tangent function, we choose a range close to the origin that allows all real numbers to have function values. The interval (-π/2, π/2) satisfies this requirement (Fig. 3).
The notation arcsin x arises because the function value, y, is the length of an arc on the unit circle for which the sine is x. Either of the two kinds of notation above can be read “the inverse sine of x” or “the arcsine of x” or “the number (or angle) whose sine is x.”
The notation sin-1 x is not exponential notation.
It does not mean 1sin x!
The graphs of the inverse trigonometric functions are as follows.
The following diagrams show the restricted ranges for the inverse trigonometric functions on a unit circle. Compare these graphs with the graphs on the preceding page. The ranges of these functions should be memorized. The missing endpoints in the graph of the arctangent function indicate inputs that are not in the domain of the original function.
Find each of the following function values. For parts (d) and (e), see the preceding Connecting the Concepts for the restricted ranges.
a) sin-1 √22
b) cos-1 (-12)
c) tan-1 (-√33)
d) cot-1 0
e) sec-1 (-√2)
a) Another way to state “find sin-1 √2/2” is to say “find β such that β=√2/2.” In the restricted range [-π/2, π/2], the only number with a sine of √2/2 is π/4. Thus, sin-1 (√2/2)=π/4, or 45°. (See Fig. 4 below.)
b) The only number with a cosine of -12 in the restricted range [0, π] is 2π/3. Thus, cos-1 (-12)=2π/3, or 120°. (See Fig. 5 above.)
c) The only number in the restricted range (-π/2, π/2) with a tangent of -√3/3 is -π/6. Thus, tan-1 (-√3/3) is -π/6, or -30°. (See Fig. 6 at left.)
d) The only number in the restricted range (0, π) with a cotangent of 0 is π/2. Thus, cot-1 1=π/2, or 90°.
e) The only number in the restricted range [0, π/2)∪(π/2, π] with a secant of -√2 is 3π/4. Thus, sec-1(-√2)=3π/4, or 135°.
Now Try Exercises 1 and 11.
We can also use a calculator to find inverse trigonometric function values. The cosecant, the secant, and the cotangent functions are the reciprocals of the sine, the cosine, and the tangent functions, respectively. Thus we have
On most graphing calculators, we can find inverse function values in either radians or degrees simply by selecting the appropriate mode. The keystrokes involved in finding inverse function values vary with the calculator. Be sure to read the instructions for the particular calculator that you are using.
Approximate each of the following function values in both radians and degrees. Round radian measure to four decimal places and degree measure to the nearest tenth of a degree.
a) cos-1 (-0.2689)
b) tan-1 (-0.2623)
c) sin-1 0.20345
d) cos-1 1.318
e) csc-1 8.205
FUNCTION VALUE | MODE | READOUT | ROUNDED |
---|---|---|---|
a) cos-1 (-0.2689) |
Radian Degree |
|
1.8430 105.6° |
b) tan-1 (-0.2623) |
Radian Degree |
|
-0.2565 -14.7∘ |
c) sin-1 0.20345 |
Radian Degree |
|
Radian 0.2049 11.7° |
d) cos-1 1.318 |
Radian Degree |
|
|
The value 1.318 is not in [-1,1], the domain of the arccosine function. |
|||
e) csc-1 8.205=sin-1 (1/8.205) |
Radian Degree |
|
0.1222 7.0° |
Now Try Exercises 21, 25 and 27.
Various compositions of trigonometric functions and their inverses often occur in practice. For example, we might want to try to simplify an expression such as
In the expression on the left, we are finding “the sine of a number whose sine is x.” Recall from Section 5.1 that if a function f has an inverse that is also a function, then
and
Thus, if f(x)=sin x and f-1(x)=sin-1 x, then
which is any number on the interval [-1,1]. Similar results hold for the other trigonometric functions.
Simplify each of the following.
a) cos (cos-1 √32)
b) sin (sin-1 1.8)
a) Since √3/2 is in [-1,1], the domain of cos-1, it follows that
b) Since 1.8 is not in [-1,1], the domain of sin-1, we cannot evaluate this expression. We know that there is no number with a sine of 1.8. Because we cannot find sin-1 1.8, we state that sin (sin-1 1.8) does not exist.
Now Try Exercise 37.
Now let’s consider an expression like sin-1 (sin x). We might also suspect that this is equal to x for any x in the domain of sin x, but this is not true unless x is in the range of the sin-1 function. Note that in order to define sin-1, we had to restrict the domain of the sine function. In doing so, we restricted the range of the inverse sine function. Thus,
Similar results hold for the other trigonometric functions.
Simplify each of the following.
a) tan-1 (tan π6)
b) sin-1 (sin 3π4)
a) Because π/6 is in (-π/2, π/2), the range of the tan-1 function, we can use tan-1 (tan x)=x. Thus,
b) Note that 3π/4 is not in [-π/2, π/2], the range of the sin-1 function. Thus we cannot apply sin-1 (sin x)=x. Instead we first find sin (3π/4), which is √2/2, and substitute:
Now Try Exercises 41 and 43.
Now we find some other function compositions.
Simplify each of the following.
a) sin [tan-1 (-1)]
b) cos-1 (sin π2)
a) We know that tan-1 (-1) is the number (or angle) θ in (-π/2, π/2) whose tangent is -1. That is, tan θ=-1. Thus, θ=-π/4 and
b) cos-1 (sin π2)=cos-1 (1)=0sin π2 = 1
Now Try Exercises 47 and 49.
Without using a calculator, we cannot find sin-1 35. However, we can still evaluate the entire expression by sketching a reference triangle. We are looking for angle θ such that sin-1 35=θ, or sin θ=35. Since sin-1 is defined in [-π/2, π/2] and 35>0, we know that θ is in quadrant I. We sketch a reference right triangle, as shown at left. The angle θ in this triangle is an angle whose sine is 35. We wish to find the cosine of this angle. Since the triangle is a right triangle, we can find the length of the base, b. It is 4. Thus we know that cos θ=b/5, or 45. Therefore,
Find sin (cot-1 x2).
Since cot-1 is defined in (0, π), we consider quadrants I and II. We draw right triangles, as shown at left, whose legs have lengths x and 2, so that cot θ=x/2.
In each, we find the length of the hypotenuse and then read off the sine ratio. We get
Now Try Exercise 55.
In the following example, we use a sum identity to evaluate an expression.
Evaluate:
Since sin-1 12 and cos-1 513 are both angles, the expression is the sine of a sum of two angles, so we use the identity
Thus,
Now since sin-1 12=π/6, cos (sin-1 12) simplifies to cos π/6, or √3/2. We can illustrate this with a reference triangle in quadrant I. (See Fig. 1.)
To find sin (cos-1 513), we use a reference triangle in quadrant I and determine that the sine of the angle whose cosine is 513 is 1213. (See Fig. 2.)
Our expression now simplifies to
Thus,
Now Try Exercise 63.
Find each of the following exactly in radians and degrees.
1. sin-1 (-√32)
2. cos-1 12
3. tan-1 1
4. sin-1 0
5. cos-1 √22
6. sec-1 √2
7. tan-1 0
8. tan-1 √33
9. cos-1 √32
10. cot-1 (-√33)
11. csc-1 2
12. sin-1 12
13. cot-1 (-√3)
14. tan-1 (-1)
15. sin-1 (-12)
16. cos-1 (-√22)
17. cos-1 0
18. sin-1 √32
19. sec-1 2
20. csc-1 (-1)
Use a calculator to find each of the following in radians, rounded to four decimal places, and in degrees, rounded to the nearest tenth of a degree.
21. tan-1 0.3673
22. cos-1 (-0.2935)
23. sin-1 0.9613
24. sin-1 (-0.6199)
25. cos-1 (-0.9810)
26. tan-1 158
27. csc-1 (-6.2774)
28. sec-1 1.1677
29. tan-1 (1.091)
30. cot-1 1.265
31. sin-1 (-0.8192)
32. cos-1 (-0.2716)
33. State the domains of the inverse sine, the inverse cosine, and the inverse tangent functions.
34. State the ranges of the inverse sine, the inverse cosine, and the inverse tangent functions.
Evaluate.
37. sin (sin-1 0.3)
38. tan [tan-1 (-4.2)]
39. cos-1 [cos (-π4)]
40. sin-1 (sin 2π3)
41. sin-1 (sin π5)
42. cot-1 (cot 2π3)
43. tan-1 (tan 2π3)
44. cos-1 (cos π7)
45. sin (tan-1 √33)
46. cos (sin-1 √32)
47. tan (cos-1 √22)
48. cos-1 (sin π)
49. sin-1 (cos π6)
50. sin-1 [tan (-π4)]
51. tan (sin-1 0.1)
52. cos (tan-1 √34)
53. sin-1 (sin 7π6)
54. tan-1 [tan (-3π4)]
Find each of the following.
55. sin (tan-1 a3)
56. tan (cos-1 3x)
57. cot (sin-1 pq)
58. sin (cos-1 x)
59. tan (sin-1 p√p2+9)
60. tan (12 sin-1 12)
61. cos (12 sin-1 √32)
62. sin (2 cos-1 35)
Evaluate.
63. cos (sin-1 √22+cos-1 35)
64. sin (sin-1 12+cos-1 35)
65. sin (sin-1 x+cos-1 y)
66. cos (sin-1 x-cos-1 y)
67. sin (sin-1 0.6032+cos-1 0.4621)
68. cos (sin-1 0.7325-cos-1 0.4838)
In each of Exercises 69–76, fill in the blank with the correct term. Some of the given choices will not be used.
linear speed
angular speed
angle of elevation
angle of depression
complementary
supplementary
similar
congruent
circular
periodic
period
amplitude
quadrantal
radian measure
69. A function f is said to be if there exists a positive constant p such that f(s+p)=f(s) for all s in the domain of f.[6.5]
70. The of a rotation is the ratio of the distance s traveled by a point at a radius r from the center of rotation to the length of the radius r.[6.4]
71. Triangles are if their corresponding angles have the same measure.[6.1]
72. The angle between the horizontal and a line of sight below the horizontal is called a(n) .[6.2]
73. is the amount of rotation per unit of time.[6.4]
74. Two positive angles are if their sum is 180°.[6.3]
75. The of a periodic function is one-half of the distance between its maximum and minimum function values.[6.5]
76. Trigonometric functions with domains composed of real numbers are called functions. [6.5]
Prove the identity.
77. sin-1 x+cos-1 x=π2
78. tan-1 x=sin-1 x√x2+1
79. sin-1 x=cos-1 √1-x2, for x≥0
80. cos-1 x=tan-1 √1-x2x, for x>0
81. Height of a Mural. An art student’s eye is at a point A, looking at a mural of height h, with the bottom of the mural y feet above the eye (see the figure). The eye is x feet from the wall. Write an expression for θ in terms of x, y, and h. Then evaluate the expression when x=20 ft,y=7 ft, and h=25 ft.
82. Use a calculator to approximate the following expression:
What number does this expression seem to approximate?