Inverse Trig Functions

Abstract

Since sine and cosine are periodic, they aren't invertible on their whole domain

Sine is invertible at
- has domain and range
Cosine is invertible at
- has domain and range
Tangent is invertible on
- has domain and range
Secant is invertible on
- has domain and range

First thing to note is that periodic functions do not have inverses, since they are not injective. Instead, the inverse trigonometric functions have restrictions imposed on their domains.

Inverse Sine

The function is invertible at

Arcsine Interval.png

Definition

Note

is true for only
is true for only

Example

How did we get the second value? We can use the CAST rule of definition of sine to turn to by subtracting by

Important according to Prof. JINTAO DENG

What does the graph of look like?

left = -15; right = 15;
bottom = -5; top = 5;
---
y = \sin^{-1}(\sin(x)) \{-\frac{\pi}{2} \le x \le \frac{\pi}{2}\}

Since the function is periodic, with period

left = -15; right = 15;
bottom = -5; top = 5;
---
y = \sin^{-1}(\sin(x)) \{-\frac{\pi}{2} \le x \le \frac{\pi}{2}\}
y = \sin^{-1}(\sin(x)) \{-\frac{5\pi}{2} \le x \le -\frac{3\pi}{2}\}
y = \sin^{-1}(\sin(x)) \{\frac{3\pi}{2} \le x \le \frac{5\pi}{2}\}
y = \sin^{-1}(\sin(x)) \{-\frac{9\pi}{2} \le x \le -\frac{7\pi}{2}\}
y = \sin^{-1}(\sin(x)) \{\frac{7\pi}{2}\ \le x \le \frac{9\pi}{2}\}

We know that from the symmetry identities. Because the cosine function is even, we realize that our own function is essentially an even function shifted to the right. In other words, the graph must be symmetric about the line .

left = -15; right = 15;
bottom = -5; top = 5;
---
y = \sin^{-1}(\sin(x))

Inverse Cosine

The function is invertible at

Arccosine Inverval.png

Definition

Inverse Tangent

The function is invertible at

Arctangent Interval.png

Definition

Inverse Secant

The function is invertible at

Arcsecant Interval.png

Definition