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Why are domain restrictions necessary for inverse trig functions?

Trig functions are periodic, so domain restrictions ensure unique inverse values.

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Why are domain restrictions necessary for inverse trig functions?
Trig functions are periodic, so domain restrictions ensure unique inverse values.
What is the range of arcsine?
$[-\frac{\pi}{2}, \frac{\pi}{2}]$
What is the range of arccosine?
$[0, \pi]$
What is the range of arctangent?
$(-\frac{\pi}{2}, \frac{\pi}{2})$
Explain how the unit circle helps evaluate inverse trig functions.
It visually represents angles and their corresponding sine, cosine, and tangent values, making it easy to find the angle for a given ratio.
Describe the relationship between trig functions and inverse trig functions.
Inverse trig functions 'undo' what trig functions do. If $\sin(x) = y$, then $\arcsin(y) = x$.
How do you evaluate $\arcsin(\frac{1}{2})$?
Find the angle within the range of arcsine ($[-\frac{\pi}{2}, \frac{\pi}{2}]$) whose sine is $\frac{1}{2}$. Answer: $\frac{\pi}{6}$.
How do you evaluate $\arccos(0)$?
Find the angle within the range of arccosine ($[0, \pi]$) whose cosine is 0. Answer: $\frac{\pi}{2}$.
How do you evaluate $\arctan(1)$?
Find the angle within the range of arctangent $(-\frac{\pi}{2}, \frac{\pi}{2})$ whose tangent is 1. Answer: $\frac{\pi}{4}$.
How to solve $\sin(\arccos(x))$?
Let $\theta = \arccos(x)$. Then $\cos(\theta) = x$. Draw a right triangle, find the missing side using the Pythagorean theorem, and then find $\sin(\theta)$.
Evaluate $\tan(\arcsin(\frac{3}{5}))$
Let $\theta = \arcsin(\frac{3}{5})$. Then $\sin(\theta) = \frac{3}{5}$. Draw a right triangle, find the adjacent side using the Pythagorean theorem (which is 4), and then find $\tan(\theta) = \frac{3}{4}$.
Evaluate $\cos(\arctan(2))$
Let $\theta = \arctan(2)$. Then $\tan(\theta) = 2$. Draw a right triangle, find the hypotenuse using the Pythagorean theorem (which is $\sqrt{5}$), and then find $\cos(\theta) = \frac{1}{\sqrt{5}}$.
Define arcsine.
The inverse function of sine; finds the angle whose sine is a given value.
Define arccosine.
The inverse function of cosine; finds the angle whose cosine is a given value.
Define arctangent.
The inverse function of tangent; finds the angle whose tangent is a given value.
What does $\sin^{-1}(x)$ represent?
The angle whose sine is x.
What does $\cos^{-1}(x)$ represent?
The angle whose cosine is x.
What does $\tan^{-1}(x)$ represent?
The angle whose tangent is x.