2EInverseTrigFuncs

Inverse Trig Functions

SIne

If we take the ** inverse ** of the trig function, y = sin(x), we get a graph which is a relation but no longer a function.

In order to change y = sin(x) into a one-to-one function, so that its inverse is also a function,

math \text{we restrict the domain of sin(x) to: } \left[ -\dfrac{\pi}{2}, \;\dfrac{\pi}{2} \right] math

This specific domain is called the ** Principal Domain ** for sin(x).

Use capital "S" to indicate sine with its domain restricted to the **Principal Domain**.

Sin(x) is defined as: math . \qquad y=Sin(x), \;\; x \in \left[ -\dfrac{\pi}{2},\; \dfrac{\pi}{2} \right] \quad \text {Range } y \in [-1, \, 1] math

The Inverse Sine function y ``=`` Sin –1 (x) is the reflection of that graph across the line y ``=`` x.

This swaps the x and y values, so it also swaps the domain and range.

math . \qquad y=Sin^{-1}(x), \;\; x \in [-1, \, 1] \quad \text {Range } y \in \left[ -\dfrac{\pi}{2}, \; \dfrac{\pi}{2} \right] math


 * Notes **

math .\quad \centerdot \; Sin \left(Sin^{-1}x\right) = x, \;\; x\in \left[ -\dfrac{\pi}{2}, \;\dfrac{\pi}{2} \right] math
 * Sin –1 (x) is also called ** arcsin(x **), or ** arsin(x) **, or ** asin(x) **
 * The text we are using makes no distinction between Sin –1 (x) and sin –1 (x) {Grrr!}

Cosine

For cosine, we have to use a different **Principal Domain** to get a defined inverse function.

Cos(x) is defined as: math . \qquad y=Cos(x), \;\; x\in [0,\, \pi] \quad \text {Range } y\in [-1, \, 1] math

The inverse of Cos(x) is: math . \qquad y=Cos^{-1}, \;\; x \in [-1, \,1] \quad \text {Range } y \in [0, \, \pi] math



math .\quad \centerdot \;\; Cos \left(Cos^{-1}x\right) = x, \;\; x\in [0, \, \pi] math
 * Notes: **
 * Cos –1 (x) is also called ** arccos(x) **, or ** arcos(x) **, or ** acos(x) **

Tangent

For tangent, we have to consider the asymptotes when identifying the restricted domain to get a defined inverse function. That is why the **Principal Domain** does not include the endpoints.

Tan(x) is defined as: math \\ . \qquad y=Tan(x), \;\; x \in \left( -\dfrac{\pi}{2}, \; \dfrac{\pi}{2} \right) \quad \text {Range: } y \in R \\. \\ . \qquad \text{Asympotes } \; x = -\dfrac{\pi}{2}, \;\; x = \dfrac{\pi}{2} math

The inverse of Tan(x) is: math \\ . \qquad y=Tan^{-1}(x), \;\; x \in R \quad \text {Range: } y \in \left( -\dfrac{\pi}{2}, \; \dfrac{\pi}{2} \right) \\. \\ . \qquad \text {Asymptotes } \; y=-\dfrac{\pi}{2}, \;\; y=\dfrac{\pi}{2} math



Notice
 * the horizontal asymptotes
 * the domain covers all of R.

math .\quad \centerdot \; Tan \left(Tan^{-1}x\right) = x, \;\; x\in \left( -\dfrac{\pi}{2}, \;\dfrac{\pi}{2} \right) math
 * Notes **
 * Tan –1 (x) is also called ** arctan(x) **, or ** artan(x) **, or ** atan(x) **

CAS Calculators

In function mode, the Classpad knows about the Principal Domains, so if you sketch y = Sin –1 (x), {or any of the others}, you get the function with the correct domain and range.

Example 1

{Look carefully for capital/lower case letters} math . \qquad Cos^{-1} \left( sin \dfrac {4\pi}{3} \right) = Cos^{-1} \left( -\dfrac{\sqrt{3}}{2} \right) = \dfrac{2\pi}{3} math

{because range is [0, p ]}

math . \qquad Cos^{-1} \left( Sin \dfrac {4\pi}{3} \right) = \text{ no solution} math

This is undefined because 4 p /3 is outside the domain of Sin

Example 2

State the implied domain and the range of 3Sin –1 (2x + 1)

{The implied domain is the maximum possible domain for this function}

The domain of Sin –1 (y) is [–1, 1]

Thus the implied domain is the solution to the inequation: math \\ . \qquad -1 \leqslant 2x+1 \leqslant 1 \\. \\ . \qquad -2 \leqslant 2x \leqslant 0 \\. \\ . \qquad -1 \leqslant x \leqslant 0 math

Hence the implied domain is x Î [–1, 0]

{The output of Sin –1 (y) is multiplied by 3, so do the same to the range:} math \\ . \qquad y \in 3\left[ -\dfrac{\pi}{2},\dfrac{\pi}{2} \right] \\. \\ . \qquad y \in \left[ -\dfrac{3\pi}{2},\dfrac{3\pi}{2} \right] math

More examples here

**See also** www.mathsonline.com.au Y12Extension --> Functions --> Inverse Functions --> Lessons 2, 3

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