19 Section 4.3: Inverses of Trigonometric Functions
Example 1
Without using a calculator, determine the exact values of the following numerical expressions
a. [latex]\arcsin(\cos(\pi))[/latex]
c. [latex]\arctan(\cos(\frac{\pi}{2}))[/latex]
d. [latex]\arcsin(\cos(\frac{4\pi}{3}))[/latex]
e. [latex]\arcsin(\tan(\frac{7\pi}{4})[/latex]
f. [latex]\arctan(\tan\frac{2\pi}{3})[/latex]
g. [latex]\arccos(\cos(\frac{7\pi}{6}))[/latex]
Show Solution
a. [latex]\arcsin(\cos(\pi)) = \arcsin(-1) = -\frac{\pi}{2}.[/latex]
b. [latex]\arctan(\cos(\frac{\pi}{2})) = \arctan(0) = 0[/latex]
d. [latex]\arccos(\sin(\frac{4\pi}{3})) = \arccos(-\frac{\sqrt{3}}{2}) = \frac{5\pi}{6}.[/latex]
e. [latex]\arcsin(\tan(\frac{7\pi}{4})) = \arcsin(-1) = \frac{-\pi}{2}.[/latex]
f. [latex]\arctan(\tan(\frac{2\pi}{3})) = \arctan(\frac{-\sqrt{3}}{2}) = \frac{-\pi}{3}.[/latex]
h. [latex]\arccos(\cos(\frac{7\pi}{6})) = \arccos(\frac{-1}{2}) = \frac{5\pi}{6}.[/latex]
Example 2
Without using a calculator, determine the exact values of the following numerical expressions
a. [latex]\cos(\arcsin(\frac{\sqrt{3}}{2})[/latex]
b. [latex]\sin(\arccos(\frac{-1}{2}))[/latex]
c. [latex]\cos(\arctan(\frac{-1}{\sqrt{3}}))[/latex]
d. [latex]\tan(\arcsin(1))[/latex]
e. [latex]\tan(\arccos(1))[/latex]
f. [latex]\sin(\arccos(-\frac{\sqrt{2}}{2}))[/latex]
Show Solution
a. [latex]\cos(\arcsin(\frac{\sqrt{3}}{2})) = \cos(\frac{\pi}{3}) = \frac{1}{2}.[/latex]
b. [latex]\sin(\arccos(\frac{-1}{2})) = \sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2}.[/latex]
c. [latex]\cos(\arctan(\frac{-1}{\sqrt{3}})) = \cos(\frac{-\pi}{6}) = \frac{\sqrt{3}}{2}.[/latex]
d. [latex]\tan(\arcsin(1)) = \tan(\frac{\pi}{2}) =[/latex] undefined.
e. [latex]\tan(\arccos(1)) = \tan(0) = 0.[/latex]
f. [latex]\sin(\arccos(-\frac{\sqrt{2}}{2})) = \sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2}.[/latex]