Inverse Trigonometric Functions

The One Thing

Inverse trigonometric functions return the unique principal angle corresponding to a given trigonometric ratio. Their domains, ranges, and identities must be used consistently, because the same ratio may correspond to many angles outside the specified principal-value interval.

Definitions and Results

• Inverse trigonometric function: A function that returns an angle when the value of a trigonometric ratio is known; it is written as ${\sin }^{-1}x$, ${\cos }^{-1}x$, ${\tan }^{-1}x$, and similarly for the other ratios. The notation ${\sin }^{-1}x$ means inverse sine, not $1/\sin x$; the reciprocal is written $\mathrm{cosec}x$ or $1/\sin x$.

• Principal value: The unique angle selected from a specified interval as the value of an inverse trigonometric function.

• Restricted trigonometric function: A trigonometric function limited to an interval where it is one-to-one, allowing an inverse function to exist.

• ${\sin }^{-1}x$: The angle $y\in [-\pi /2,\pi /2]$ such that $\sin y=x$. Its domain is $[-1,1]$, and its range is $[-\pi /2,\pi /2]$.

• ${\cos }^{-1}x$: The angle $y\in [0,\pi ]$ such that $\cos y=x$. Its domain is $[-1,1]$, and its range is $[0,\pi ]$.

• ${\tan }^{-1}x$: The angle $y\in (-\pi /2,\pi /2)$ such that $\tan y=x$. Its domain is all real numbers, and its range is $(-\pi /2,\pi /2)$.

• ${\cot }^{-1}x$: The angle $y\in (0,\pi )$ such that $\cot y=x$. Its domain is all real numbers, and its range is $(0,\pi )$.

• ${\sec }^{-1}x$: The principal angle $y$, usually taken in $[0,\pi ]$ excluding $\pi /2$, such that $\sec y=x$. Its domain is

$$(-\infty ,-1]\cup [1,\infty ).$$

• ${\mathrm{cosec}}^{-1}x$: The principal angle $y$, usually taken in $[-\pi /2,\pi /2]$ excluding $0$, such that $\mathrm{cosec}y=x$. Its domain is

$$(-\infty ,-1]\cup [1,\infty ).$$

• Composition of inverse functions: Expressions such as $\sin ({\sin }^{-1}x)$ or ${\sin }^{-1}(\sin x)$. Their simplification depends on whether the input angle lies in the relevant principal-value range.

• Basic compositions:

$$\sin ({\sin }^{-1}x)=x,-1\le x\le 1,$$ $$\cos ({\cos }^{-1}x)=x,-1\le x\le 1,$$ $$\tan ({\tan }^{-1}x)=x,x\in \mathbb{R}.$$

• Reverse compositions:

$${\sin }^{-1}(\sin x)=x$$ only when $x\in [-\pi /2,\pi /2]$; $${\cos }^{-1}(\cos x)=x$$ only when $x\in [0,\pi ]$; $${\tan }^{-1}(\tan x)=x$$ only when $x\in (-\pi /2,\pi /2)$.

• Oddness and related identities:

$${\sin }^{-1}(-x)=-{\sin }^{-1}x,$$ $${\tan }^{-1}(-x)=-{\tan }^{-1}x,$$ $${\cot }^{-1}(-x)=\pi -{\cot }^{-1}x,$$ where ${\cot }^{-1}x$ has range $(0,\pi )$.

• Complementary-angle identities:

$${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2},-1\le x\le 1,$$ $${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2},$$ for every real $x$, when ${\cot }^{-1}x$ has range $(0,\pi )$.

• Addition and subtraction of inverse tangents:

$${\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\left(\frac{x+y}{1-xy}\right),$$ for suitable real $x$ and $y$, with an additional adjustment of $\pi$ when required by the principal-value range.

$${\tan }^{-1}x-{\tan }^{-1}y={\tan }^{-1}\left(\frac{x-y}{1+xy}\right),$$ for suitable real $x$ and $y$, with attention to the principal-value range.

• Double-angle identity:

$$2{\tan }^{-1}x={\tan }^{-1}\left(\frac{2x}{1-x^2}\right),$$ only when the resulting angle is interpreted according to the correct principal range.

• Angle units: All angles may be expressed in radians or degrees, but the unit must remain consistent throughout a calculation.

• Graphs and inverses: The graph, domain, and range of an inverse trigonometric function are related to those of the restricted original function through reflection in the line $y=x$.

Worked Methods

1. Establishing an inverse function

• Identify the original trigonometric function.
• Restrict it to an interval on which it is one-to-one.
• Define the inverse using the corresponding principal-value range.
• Check that the input lies in the inverse function’s domain.

For example, $\sin x$ is restricted to $$[-\pi /2,\pi /2],$$ where it is one-to-one. Its inverse is therefore ${\sin }^{-1}x$, with domain $[-1,1]$ and range $[-\pi /2,\pi /2]$.

Similarly, $\cos x$ is restricted to $[0,\pi ]$, and $\tan x$ is restricted to $(-\pi /2,\pi /2)$.

2. Simplifying a direct composition

For an expression such as $$\sin ({\sin }^{-1}x),$$ use the inverse relationship directly: $$\sin ({\sin }^{-1}x)=x,-1\le x\le 1.$$

The same method gives $$\cos ({\cos }^{-1}x)=x,-1\le x\le 1,$$ and $$\tan ({\tan }^{-1}x)=x,x\in \mathbb{R}.$$

3. Simplifying a reverse composition

For an expression such as $${\sin }^{-1}(\sin x),$$ first determine whether $x$ lies in the principal range of ${\sin }^{-1}$: $$[-\pi /2,\pi /2].$$

• If $x\in [-\pi /2,\pi /2]$, then

$${\sin }^{-1}(\sin x)=x.$$

• If $x$ lies outside this interval, use periodicity and symmetry to find the equivalent angle in $[-\pi /2,\pi /2]$.

The same procedure applies to $${\cos }^{-1}(\cos x),$$ using the principal range $[0,\pi ]$, and to $${\tan }^{-1}(\tan x),$$ using the principal range $(-\pi /2,\pi /2)$.

4. Using complementary-angle identities

To simplify $${\sin }^{-1}x+{\cos }^{-1}x,$$ recognise that the two principal angles are complementary: $${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2},-1\le x\le 1.$$

Likewise, under the range convention ${\cot }^{-1}x\in (0,\pi )$, $${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}.$$

5. Using inverse-tangent addition and subtraction

For a sum, $${\tan }^{-1}x+{\tan }^{-1}y,$$ apply $${\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\left(\frac{x+y}{1-xy}\right),$$ then check whether the resulting angle lies in the correct principal range. If not, adjust by $\pi$.

For a difference, $${\tan }^{-1}x-{\tan }^{-1}y,$$ apply $${\tan }^{-1}x-{\tan }^{-1}y={\tan }^{-1}\left(\frac{x-y}{1+xy}\right),$$ again checking the principal-value condition.

6. Using the double-angle identity

For $$2{\tan }^{-1}x,$$ use $$2{\tan }^{-1}x={\tan }^{-1}\left(\frac{2x}{1-x^2}\right),$$ but determine the correct principal interpretation of the resulting angle. The formula cannot be applied without considering the angle range.

Where It Goes Wrong

• Treating ${\sin }^{-1}x$ as $1/\sin x$; the former is inverse sine, whereas the reciprocal is $\mathrm{cosec}x$ or $1/\sin x$.
• Applying ${\sin }^{-1}(\sin x)=x$, ${\cos }^{-1}(\cos x)=x$, or ${\tan }^{-1}(\tan x)=x$ without checking whether $x$ lies in the relevant principal-value range.
• Forgetting the domain restrictions $[-1,1]$ for ${\sin }^{-1}x$ and ${\cos }^{-1}x$, or the domain restriction $(-\infty ,-1]\cup [1,\infty )$ for ${\sec }^{-1}x$ and ${\mathrm{cosec}}^{-1}x$.
• Using the inverse-tangent addition, subtraction, or double-angle identities without checking whether an additional adjustment of $\pi$ is required.
• Applying ${\cot }^{-1}(-x)=\pi -{\cot }^{-1}x$ while using a different range convention from $(0,\pi )$.
• Mixing radians and degrees during a calculation; the angle unit must remain consistent.

What Gets Asked

This material supports questions requiring students to:

• State the domains, ranges, and principal values of ${\sin }^{-1}x$, ${\cos }^{-1}x$, ${\tan }^{-1}x$, ${\cot }^{-1}x$, ${\sec }^{-1}x$, and ${\mathrm{cosec}}^{-1}x$.
• Explain why a trigonometric function must be restricted to a one-to-one interval before it can have an inverse.
• Simplify compositions such as

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

• Use oddness, complementary-angle, inverse-tangent addition, inverse-tangent subtraction, and double-angle identities.
• Determine when an inverse-trigonometric identity requires an adjustment of $\pi$.
• Use periodicity and symmetry to locate the principal angle corresponding to a non-principal input angle.
• Explain the relationship between the graph of a restricted trigonometric function and the graph of its inverse through reflection in $y=x$.
• Maintain consistent angle units and distinguish inverse notation from reciprocal notation.