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6.6 Inverse Trigonometric
Functions
Inverse Trigonometric Functions
You can see from Figure 1 that the sine function y = sin x is not one-to-one (use the Horizontal Line Test).
Figure 1
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Inverse Trigonometric Functions
But the function f(x) = sin x, –π/2 ≤ x ≤ π/2 is one-to-one (see Figure 2).The inverse function of this restricted sine
function f exists and is denoted by sin–1 or arcsin. It is called the inverse sine function or the arcsine function.
Figure 2
y = sin x,
Inverse Trigonometric Functions
Since the definition of an inverse function says that f–1(x) = y f(y) = x
we have
Thus, if –1 ≤ x ≤ 1, sin–1x is the number between –π/2 and π/2 whose sine is x.
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Example 1
Evaluate (a) sin–1 and (b) tan(arcsin ).
Solution:
(a) We have
because sin(π/6) = and π/6 lies between –π/2 and π/2.
Example 1 – Solution
(b) Let θ = arcsin , so sin θ = . Then we can draw a right triangle with angle θ as in Figure 3 and deduce from the Pythagorean Theorem that the third side has length
This enables us to read from the triangle that
Figure 3
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Inverse Trigonometric Functions
The cancellation equations for inverse functions become, in this case,
Inverse Trigonometric Functions
The inverse sine function, sin–1, has domain [–1, 1] and range [–π/2, π/2], and its graph, shown in Figure 4, is
obtained from that of the restricted sine function (Figure 2) by reflection about the line y = x.
Figure 4
y = sin–1x = arcsin x
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Inverse Trigonometric Functions
We know that the sine function f is continuous, so the
inverse sine function is also continuous. The sine function is differentiable, so the inverse sine function is also
differentiable.
Let y = sin–1x. Then sin y = x and –π/2 ≤ y ≤ π/2.
Differentiating sin y = x implicitly with respect to x, we obtain
and
Inverse Trigonometric Functions
Now cos y ≥ 0 since –π/2 ≤ y ≤ π/2, so
Therefore
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Example 2
If f(x) = sin–1(x2 – 1), find (a) the domain of f, (b) f′(x), and (c) the domain of f′.
Solution:
(a) Since the domain of the inverse sine function is [–1, 1], the domain of f is
{x | –1 ≤ x2 – 1 ≤ 1} = {x | 0 ≤ x2 ≤ 2}
Example 2 – Solution
(b) Combining Formula 3 with the Chain Rule, we have
(c) The domain of f′ is
{x | –1 < x2 – 1 < 1} = {x | 0 < x2 < 2}
cont’d
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Inverse Trigonometric Functions
The inverse cosine function is handled similarly. The restricted cosine function f(x) = cos x, 0 ≤ x ≤ π, is
one-to-one (see Figure 6) and so it has an inverse function denoted by cos–1 or arccos.
Figure 6
y = cos x, 0 ≤ x ≤ π
Inverse Trigonometric Functions
The cancellation equations are
The inverse cosine function, cos–1, has domain [–1, 1] and range [0, π] and is a continuous function whose graph is shown in Figure 7.
Figure 7
y = cos–1x = arccos x
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Inverse Trigonometric Functions
Its derivative is given by
The tangent function can be made one-to-one by restricting it to the interval (–π/2 , π/2).
Thus the inverse tangent function is defined as the inverse of the function f(x) = tan x, –π/2 < x < π/2.
Inverse Trigonometric Functions
It is denoted by tan–1 or arctan. (See Figure 8.)
Figure 8
y = tan x, < x <
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Example 3
Simplify the expression cos(tan–1x).
Solution1:
Let y = tan–1x. Then tan y = x and –π/2 < y < π/2. We want to find cos y but, since tan y is known, it is easier to find sec y first:
sec2y = 1 + tan2y
= 1 + x2
Thus
(since sec y > 0 for –π/2 < y < π/2)
Example 3 – Solution 2
Instead of using trigonometric identities as in Solution 1, it is perhaps easier to use a diagram.
If y = tan–1x, then tan y = x, and we can read from Figure 9 (which illustrates the case y > 0) that
cont’d
Figure 9
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Inverse Trigonometric Functions
The inverse tangent function, tan–1x = arctan, has domain and range (–π/2, π/2). Its graph is shown in Figure 10.
Figure 10
y = tan–1x = arctan x
Inverse Trigonometric Functions
We know that
and
and so the lines x = ±π/2 are vertical asymptotes of the graph of tan.
Since the graph of tan–1 is obtained by reflecting the graph of the restricted tangent function about the line y = x, it
follows that the lines y = π/2 and y = –π/2 are horizontal asymptotes of the graph of tan–1.
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Inverse Trigonometric Functions
This fact is expressed by the following limits:
Example 4
Evaluate arctan Solution:
If we let t = 1/(x – 2), we know that t → as x → 2+. Therefore, by the first equation in (8), we have
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Inverse Trigonometric Functions
Because tan is differentiable, tan–1 is also differentiable. To find its derivative, we let y = tan–1x.
Then tan y = x. Differentiating this latter equation implicitly with respect to x, we have
and so
Inverse Trigonometric Functions
The remaining inverse trigonometric functions are not used as frequently and are summarized here.
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Inverse Trigonometric Functions
We collect in Table 11 the differentiation formulas for all of the inverse trigonometric functions.
Inverse Trigonometric Functions
Each of these formulas can be combined with the Chain Rule. For instance, if u is a differentiable function of x, then
and
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Example 5
Differentiate (a) y = and (b) f(x) = x arctan . Solution:
(a)
(b)
Inverse Trigonometric Functions
29
Example 7
Find
Solution:
If we write
then the integral resembles Equation 12 and the substitution u = 2x is suggested.
Example 7 – Solution
This gives du = 2 dx, so dx = du /2. When x = 0, u = 0;
when x = , u = . So
cont’d
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Inverse Trigonometric Functions
Example 9
Find
Solution:
We substitute u = x2 because then du = 2x dx and we can use Equation 14 with a = 3:
