Inverse Trigonometric Functions

The One Thing

Inverse trigonometric functions return unique principal-value angles corresponding to specified trigonometric ratios. Their identities are valid only when the relevant domain, principal range, signs, and quadrants are respected.

Definitions and Results

• Inverse trigonometric function: A function that returns an angle whose trigonometric ratio has a specified value, written as ${\sin }^{-1}x$, ${\cos }^{-1}x$, ${\tan }^{-1}x$, and so on. These are inverse functions, not reciprocals: ${\sin }^{-1}x$ denotes the inverse function of sine, whereas $1/\sin x=\mathrm{cosec}x$.
• Restricted domain: A selected interval on which a trigonometric function is one-to-one, allowing it to have an inverse.
• One-to-one function: A function in which different input values produce different output values. This property is required for defining an inverse.
• Principal value: The unique angle selected from a prescribed interval for a given value of an inverse trigonometric function.
• Domain: The set of allowed input values for an inverse trigonometric function.
• Range: The interval from which the principal-value angle is selected.
• ${\sin }^{-1}x$: The principal angle $\theta$ such that $\sin \theta =x$, where

$$-1\le x\le 1,-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}.$$

• ${\cos }^{-1}x$: The principal angle $\theta$ such that $\cos \theta =x$, where

$$-1\le x\le 1,0\le \theta \le \pi .$$

• ${\tan }^{-1}x$: The principal angle $\theta$ such that $\tan \theta =x$, where $x$ is any real number and

$$-\frac{\pi }{2}<\theta <\frac{\pi }{2}.$$

• ${\cot }^{-1}x$: The principal angle $\theta$ such that $\cot \theta =x$, commonly taken in

$$0<\theta <\pi .$$

• ${\sec }^{-1}x$: The principal angle $\theta$ such that $\sec \theta =x$, defined for

$$x\in (-\infty ,-1]\cup [1,\infty ),$$ usually with principal values in $[0,\pi ]$, excluding $\pi /2$.

• ${\mathrm{cosec}}^{-1}x$: The principal angle $\theta$ such that $\mathrm{cosec}\theta =x$, defined for

$$x\in (-\infty ,-1]\cup [1,\infty ),$$ usually with principal values in $[-\pi /2,\pi /2]$, excluding $0$.

• Basic inverse-function identities:

$$\sin ({\sin }^{-1}x)=x(-1\le x\le 1),$$ $$\cos ({\cos }^{-1}x)=x(-1\le x\le 1),$$ $$\tan ({\tan }^{-1}x)=x\text{for every real }x.$$

• Composition identities with restrictions:

$${\sin }^{-1}(\sin x)=x\text{only if }x\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right],$$ $${\cos }^{-1}(\cos x)=x\text{only if }x\in [0,\pi ],$$ $${\tan }^{-1}(\tan x)=x\text{only if }x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right).$$

• Complementary inverse functions:

$${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}(-1\le x\le 1),$$ $${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}\text{for every real }x,$$ using the standard principal ranges.

• Odd and complementary symmetry:

$${\sin }^{-1}(-x)=-{\sin }^{-1}x,$$ $${\tan }^{-1}(-x)=-{\tan }^{-1}x,$$ $${\cos }^{-1}(-x)=\pi -{\cos }^{-1}x.$$

• Addition formula:

$${\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\left(\frac{x+y}{1-xy}\right)$$ when $xy<1$, with a suitable adjustment by $\pi$ when required by the quadrant.

• Subtraction formula:

$${\tan }^{-1}x-{\tan }^{-1}y={\tan }^{-1}\left(\frac{x-y}{1+xy}\right)$$ when $xy>-1$, with a suitable adjustment by $\pi$ when required.

• Double-angle formula:

$$2{\tan }^{-1}x={\tan }^{-1}\left(\frac{2x}{1-x^2}\right)$$ only when the resulting principal-value interval and quadrant are handled correctly.

• Graphical relationship: The graph of an inverse trigonometric function is the reflection of the graph of its corresponding restricted trigonometric function in the line $y=x$.

Worked Methods

1. Defining an inverse by restricting the domain

• Select an interval on which the trigonometric function is one-to-one.
• Use that restricted function to define a single-valued inverse.
• Select the corresponding principal-value range.

For example, sine is restricted to $$\left[-\frac{\pi }{2},\frac{\pi }{2}\right].$$ Thus, ${\sin }^{-1}x$ is the unique angle $\theta$ in this interval satisfying $$\sin \theta =x,-1\le x\le 1.$$

Similarly, cosine uses the interval $[0,\pi ]$, and tangent uses $$\left(-\frac{\pi }{2},\frac{\pi }{2}\right).$$

2. Evaluating a direct inverse-function composition

• Check that the input belongs to the domain of the inverse function.
• Use the corresponding identity.

For example: $$\sin ({\sin }^{-1}x)=x\text{for }-1\le x\le 1.$$

Likewise, $$\cos ({\cos }^{-1}x)=x\text{for }-1\le x\le 1,$$ and $$\tan ({\tan }^{-1}x)=x\text{for every real }x.$$

3. Simplifying an inverse-after-trigonometric expression

• Identify the principal range of the inverse function.
• Determine whether the original angle lies in that range.
• If it does, the expression simplifies directly; otherwise, determine the equivalent principal-value angle.

For example: $${\sin }^{-1}(\sin x)=x$$ only when $$x\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right].$$

Similarly, $${\cos }^{-1}(\cos x)=x$$ only when $x\in [0,\pi ]$, and $${\tan }^{-1}(\tan x)=x$$ only when $$x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right].$$

4. Using a right triangle for ${\tan }^{-1}x$

• Let

$$\theta ={\tan }^{-1}x.$$

• Then

$$\tan \theta =x.$$

• Represent $x$ as a ratio using a right triangle with opposite side $x$ and adjacent side $1$.
• Find the hypotenuse using the Pythagorean theorem:

$$\sqrt{x^2+1}.$$

• Obtain the remaining trigonometric ratios, subject to sign and quadrant considerations.

Thus, $$\sin \theta =\frac{x}{\sqrt{x^2+1}},\cos \theta =\frac{1}{\sqrt{x^2+1}},$$ when the chosen quadrant makes both signs positive.

5. Applying the inverse-tangent addition formula

• Identify $x$ and $y$.
• Check the condition $xy<1$.
• Apply

$${\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\left(\frac{x+y}{1-xy}\right).$$

• Check the quadrant of the original sum.
• Add or subtract $\pi$ if the principal-value expression alone lies in the wrong quadrant.

6. Applying the inverse-tangent subtraction formula

• Identify $x$ and $y$.
• Check the condition $xy>-1$.
• Apply

$${\tan }^{-1}x-{\tan }^{-1}y={\tan }^{-1}\left(\frac{x-y}{1+xy}\right).$$

• Check the quadrant of the original difference.
• Make a suitable adjustment by $\pi$ when required.

7. Applying the double-angle formula

• Begin with $2{\tan }^{-1}x$.
• Use

$$2{\tan }^{-1}x={\tan }^{-1}\left(\frac{2x}{1-x^2}\right).$$

• Determine the quadrant and principal-value interval of the left-hand side.
• Correct the result by an appropriate multiple of $\pi$ if the direct ${\tan }^{-1}$ value is not in the correct principal interval.

Where It Goes Wrong

• Treating ${\sin }^{-1}x$, ${\cos }^{-1}x$, or ${\tan }^{-1}x$ as reciprocals rather than inverse functions; $1/\sin x=\mathrm{cosec}x$, not ${\sin }^{-1}x$.
• Simplifying ${\sin }^{-1}(\sin x)$, ${\cos }^{-1}(\cos x)$, or ${\tan }^{-1}(\tan x)$ to $x$ without checking the relevant principal-value range.
• Forgetting the domain restrictions $-1\le x\le 1$ for ${\sin }^{-1}x$ and ${\cos }^{-1}x$, or the domain $x\le -1$ or $x\ge 1$ for ${\sec }^{-1}x$ and ${\mathrm{cosec}}^{-1}x$.
• Applying the inverse-tangent addition or subtraction formula without checking $xy<1$ or $xy>-1$, respectively.
• Omitting the required adjustment by $\pi$ in inverse-tangent addition, subtraction, or double-angle expressions.
• Ignoring signs and quadrants when using a right triangle or determining a principal value.

What Gets Asked

This material supports questions requiring students to:

• State the domains and principal ranges of ${\sin }^{-1}x$, ${\cos }^{-1}x$, ${\tan }^{-1}x$, ${\cot }^{-1}x$, ${\sec }^{-1}x$, and ${\mathrm{cosec}}^{-1}x$.
• Explain why a restricted domain and a one-to-one trigonometric function are necessary for defining an inverse.
• Evaluate direct compositions such as $\sin ({\sin }^{-1}x)$, $\cos ({\cos }^{-1}x)$, and $\tan ({\tan }^{-1}x)$.
• Determine when expressions such as ${\sin }^{-1}(\sin x)$, ${\cos }^{-1}(\cos x)$, and ${\tan }^{-1}(\tan x)$ equal $x$.
• Simplify expressions using

$${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$$ and $${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}.$$

• Apply the sign identities for ${\sin }^{-1}(-x)$, ${\tan }^{-1}(-x)$, and ${\cos }^{-1}(-x)$.
• Use inverse-tangent addition, subtraction, and double-angle formulas with the required conditions and quadrant adjustments.
• Construct a right triangle from $\theta ={\tan }^{-1}x$ and derive the other trigonometric ratios.
• Interpret inverse-trigonometric graphs as reflections of restricted trigonometric graphs in $y=x$.