Topic 3.9 Notes – Inverse Trigonometric Functions

Restricted domains that make trig invertible

Trig functions take an angle and return a ratio value. Inverse trig functions reverse that. They take a ratio value and return an angle.

• $\arcsin x$ or $\sin^{-1}x$
• $\arccos x$ or $\cos^{-1}x$
• $\arctan x$ or $\tan^{-1}x$

That $-1$ means inverse function, not reciprocal.

• $\sin^{-1}x \neq \frac{1}{\sin x}$
• The reciprocal of sine is cosecant, $\csc x$
• Likewise, reciprocals are secant and cotangent, not arccos or arctan

One more idea matters right away. An inverse function only exists if the original function is one-to-one, which means it passes the horizontal line test. Full sine, cosine, and tangent fail that because they repeat.

The Three Inverse Trig Functions and Their Restricted Domains

The standard restrictions are

$$\sin x \text{ on } \left[-\frac{\pi}{2},\frac{\pi}{2}\right], \quad \cos x \text{ on } [0,\pi], \quad \tan x \text{ on } \left(-\frac{\pi}{2},\frac{\pi}{2}\right)$$

These are chosen because each restricted trig function becomes one-to-one and still gives all outputs it needs. On the unit circle, these intervals pick out the angle values the inverse functions are allowed to return.

Unit circle reference angles

Arcsine

$y=\arcsin x$ means $\sin y=x$ and $-\frac{\pi}{2}\le y\le \frac{\pi}{2}$.

• domain $[-1,1]$
• range $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$
• increasing
• $\arcsin(-1)=-\frac{\pi}{2}$, $\arcsin(0)=0$, $\arcsin(1)=\frac{\pi}{2}$

Its outputs are only in Quadrant I, Quadrant IV, or on the axes.

Arccosine

$y=\arccos x$ means $\cos y=x$ and $0\le y\le \pi$.

• domain $[-1,1]$
• range $[0,\pi]$
• decreasing
• $\arccos(1)=0$, $\arccos(0)=\frac{\pi}{2}$, $\arccos(-1)=\pi$

Its outputs are never negative.

Arctangent

$y=\arctan x$ means $\tan y=x$ and $-\frac{\pi}{2}< y< \frac{\pi}{2}$.

• domain all real numbers
• range $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$
• increasing
• $\arctan(0)=0$

It never outputs $\pm \frac{\pi}{2}$.

Constructing the Inverse Analytically and Graphically

Analytically, the pattern is always the same:

1. Write the restricted trig function, like $y=\sin x$ on $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$.
2. Switch $x$ and $y$, so $x=\sin y$.
3. Rename with inverse notation, so $y=\arcsin x$.

Domain and range swap.

Graphically, inverse graphs are reflections across $y=x$.

Know these point swaps:

• sine
  • $(-\frac{\pi}{2},-1),(0,0),(\frac{\pi}{2},1)$ ↔ $(-1,-\frac{\pi}{2}),(0,0),(1,\frac{\pi}{2})$
• cosine
  • $(0,1),(\frac{\pi}{2},0),(\pi,-1)$ ↔ $(1,0),(0,\frac{\pi}{2}),(-1,\pi)$
• tangent has vertical asymptotes $x=\pm \frac{\pi}{2}$, so arctan has horizontal asymptotes $y=\pm \frac{\pi}{2}$

Evaluating and Composing Inverse Trig Functions

For exact values, treat the input as a trig value, find angles with that value, then choose the one in the inverse range.

• $\arcsin\left(-\frac{\sqrt2}{2}\right)=-\frac{\pi}{4}$
• $\arccos\left(-\frac{\sqrt3}{2}\right)=\frac{5\pi}{6}$
• $\arccos\left(-\frac{1}{2}\right)=\frac{2\pi}{3}$
• $\arctan(-\sqrt3)=-\frac{\pi}{3}$

Compositions you should know cold:

• $\sin(\arcsin x)=x$ for $-1\le x\le 1$
• $\cos(\arccos x)=x$ for $-1\le x\le 1$
• $\tan(\arctan x)=x$ for all real $x$

The reverse direction is trickier because the answer must land in the inverse’s range.

• $\arcsin(\sin \frac{3\pi}{4})=\arcsin(\frac{\sqrt2}{2})=\frac{\pi}{4}$
• $\arccos(\cos(-\frac{\pi}{3}))=\arccos(\frac{1}{2})=\frac{\pi}{3}$
• $\arctan(\tan \frac{2\pi}{3})=\arctan(-\sqrt3)=-\frac{\pi}{3}$

Common Mistakes and Calculator Checks

Most errors come from forgetting the principal value.

• $\arcsin$, $\arccos$, and $\arctan$ return one standard angle, not every angle
• $\arcsin x$ and $\arccos x$ only accept inputs in $[-1,1]$
• $\arctan x$ accepts any real input
• arctan approaches $\pm \frac{\pi}{2}$ but never equals them
• calculator mode matters. In radians mode, answers look different than in degree mode

On a quiz or FRQ, always check two things before boxing your answer. Is the input allowed and is the output in the correct interval?