Trigonometry / Unit 5: Inverse Trigonometric Functions
5.6 Properties and Graphs of Inverse Trig Functions
Notes
Symmetry of the graphs gives useful rules.
- $\arcsin(-x) = -\arcsin x$ and $\arctan(-x) = -\arctan x$ (odd functions).
- $\arccos(-x) = \pi - \arccos x$.
- $\displaystyle \arcsin x + \arccos x = \frac{\pi}{2}$.
- As $x \to \infty$, $\displaystyle \arctan x \to \frac{\pi}{2}$, so $y = \arctan x$ has horizontal asymptotes $\displaystyle y = \pm\frac{\pi}{2}$.
Practice
What is $\displaystyle \arccos\left(-\frac{1}{2}\right)$?
$\displaystyle \arccos\left(-\frac{1}{2}\right) = \pi - \arccos\frac{1}{2} = \pi - \frac{\pi}{3}$.
Suppose $\arccos a = 1.2$. What is $\arccos(-a)$?
$\arccos(-a) = \pi - \arccos a$.
As $x$ becomes very large, what value does $\arctan x$ approach?
The graph of arctangent levels off at the horizontal asymptote $\displaystyle y = \frac{\pi}{2}$.
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