Inverse trigonometric functions review

Review your knowledge of the inverse trigonometric functions, arcsin(x), arccos(x), & arctan(x).

What are the inverse trigonometric functions?

\arcsin (x)

, or

\sin^{- 1} (x)

, is the inverse of

\sin (x)

\arccos (x)

, or

\cos^{- 1} (x)

, is the inverse of

\cos (x)

\arctan (x)

, or

\tan^{- 1} (x)

, is the inverse of

\tan (x)

Range of the inverse trig functions

Radians	Degrees
$- \frac{π}{2} \leq \arcsin (θ) \leq \frac{π}{2}$ ‍	$- 90^{\circ} \leq \arcsin (θ) \leq 90^{\circ}$ ‍
$0 \leq \arccos (θ) \leq π$ ‍	$0^{\circ} \leq \arccos (θ) \leq 180^{\circ}$ ‍
$- \frac{π}{2} < \arctan (θ) < \frac{π}{2}$ ‍	$- 90^{\circ} < \arctan (θ) < 90^{\circ}$ ‍

The trigonometric functions aren't really invertible, because they have multiple inputs that have the same output. For example,

\sin (0) = \sin (π) = 0

. So what should be

\sin^{- 1} (0)

In order to define the inverse functions, we have to restrict the domain of the original functions to an interval where they are invertible. These domains determine the range of the inverse functions.

The value from the appropriate range that an inverse function returns is called the principal value of the function.

Want to learn more about arcsin(x)? Check out this video.
Want to learn more about arccos(x)? Check out this video.
Want to learn more about arctan(x)? Check out this video.

Check your understanding

Problem 1

The sine value of all options is $0.98$ ‍. Which is the principal value of $\arcsin (0.98)$ ‍?
All measures are in radians.

Want to try more problems like this? Check out this exercise.

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