Math 151 – c Lynch 1 of 4

Section 1.5 – Inverse Trigonometric Functions

Since the trigonometric function y = sin x is not one-to-one, we will restrict the function  π π  to the domain − 2 , 2 in order to define an inverse function. π π Definition. The inverse of the function f(x) = sin x, − 2 ≤ x ≤ 2 is the inverse sine function, denoted sin−1 x, defined as π π sin−1 x = y ⇐⇒ sin y = x and − ≤ y ≤ 2 2 The inverse of this restricted sine function is also called the arcsine function, denoted arcsin x.

−1 π π Note. We think of y = sin x as the angle y between − 2 and 2 with sin y = x. Example 1. Evaluate the following. √ ! 3 (a) sin−1 2

 1 (b) sin−1 − 2

  3 (c) tan arcsin − 7 Math 151 – c Lynch 1.5–Inverse Trig Functions 2 of 4

Theorem. For sin x and sin−1 x we have π π sin−1(sin x) = x for − ≤ x ≤ 2 2 sin sin−1 x
= x for − 1 ≤ x ≤ 1 π π Be careful! The first rule only holds for − 2 ≤ x ≤ 2 . Example 2. Evaluate the following. (a) sin sin−1 .63
=

 5π  (b) sin−1 sin = 13

 5π  (c) arcsin sin = 6

 17π  (d) arcsin sin = 13

Theorem. The inverse cosine function, denoted cos−1 x, is the inverse function of the function f(x) = cos x, 0 ≤ x ≤ π defined by

cos−1 x = y ⇐⇒ cos y = x and 0 ≤ y ≤ π

The domain of cos−1 x is [−1, 1] and the range is [0, π]. We also have that

cos−1(cos x) = x for 0 ≤ x ≤ π

cos cos−1 x
= x for − 1 ≤ x ≤ 1

Note. We think of y = cos−1 x as the angle y between 0 and π with cos y = x. Math 151 – c Lynch 1.5–Inverse Trig Functions 3 of 4

√ ! 2 Example 3. Evaluate cos−1 − 2

Theorem. The inverse tangent function, denoted tan−1 x, is the inverse function π π of the function f(x) = tan x, − 2 < x < 2 defined by π π tan−1 x = y ⇐⇒ tan y = x and − < y < 2 2

−1 π π
The domain of tan x is (−∞, ∞) and the range is − 2 , 2 . We also have that π π tan−1(tan x) = x for − < x < 2 2 tan tan−1 x
= x for (−∞, ∞)

−1 π π Note. We think of y = tan x as the angle y between − 2 and 2 with tan y = x. Example 4. Evaluate the following. √ (a) tan−1 − 3
=

(b) arctan (1) =

 2π  (c) cos−1 cos = 3 Math 151 – c Lynch 1.5–Inverse Trig Functions 4 of 4

 7π  (d) tan−1 tan = 6

 π  (e) arccos cos − = 3

  2 (f) cos arctan − = 3

 5 (g) sin cos−1 = 7

Example 5. Simplify the following (a) tan sin−1 x
=

(b) cos tan−1 x
=