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.
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 $\arcsin(y) = x$.
Describe the key features of the graph of $y = \arcsin(x)$.
Domain: [-1, 1], Range: $[-\frac{\pi}{2}, \frac{\pi}{2}]$, increasing function, symmetric about the origin.
Describe the key features of the graph of $y = \arccos(x)$.
Domain: [-1, 1], Range: $[0, \pi]$, decreasing function, no symmetry.
Describe the key features of the graph of $y = \arctan(x)$.
Domain: $(-\infty, \infty)$, Range: $(-\frac{\pi}{2}, \frac{\pi}{2})$, increasing function, symmetric about the origin, horizontal asymptotes at $y = \pm \frac{\pi}{2}$.
How does the range restriction of arcsin affect its graph?
The graph is bounded between $y = -\frac{\pi}{2}$ and $y = \frac{\pi}{2}$, ensuring it's a function (passes the vertical line test).
How does the range restriction of arccos affect its graph?
The graph is bounded between $y = 0$ and $y = \pi$, ensuring it's a function (passes the vertical line test).
How does the range restriction of arctan affect its graph?
The graph is bounded between $y = -\frac{\pi}{2}$ and $y = \frac{\pi}{2}$, ensuring it's a function (passes the vertical line test).
