# Exercises - Inverse Trigonometric Functions

1. Evaluate the following:

1.   $\displaystyle{\arcsin\, 1}$

2.   $\displaystyle{\arcsin\, \frac{-\sqrt{2}}{2}}$

3.   $\displaystyle{\sec\, (\textrm{arcsec}\, \sqrt{2})}$

1.   $\displaystyle{\arcsin\, (\sin\, \frac{3\pi}{4} )}$

2.   $\displaystyle{\cos\, ( \arcsin\, \frac{3}{5})}$

3.   $\displaystyle{\sin\, ( \textrm{arccot}\, \frac{x}{2})}$

2. Find the values of the following:

1.   $\displaystyle{\arcsin 0}$   $\ans{\displaystyle{0}}$

2.   $\displaystyle{\textrm{arccot}\, \frac{-\sqrt{3}}{3}}$   $\ans{\displaystyle{\frac{2\pi}{3}}}$

3.   $\displaystyle{\cot\, (\textrm{arccot}\, (-3))}$   $\ans{\displaystyle{-3}}$

4.   $\displaystyle{\cos\, (\arccos\, \frac{4}{5})}$   $\ans{\displaystyle{\frac{4}{5}}}$

5.   $\displaystyle{\textrm{arccsc}\, 2}$   $\ans{\displaystyle{\frac{\pi}{6}}}$

6.   $\displaystyle{\arccos\, (-1)}$   $\ans{\displaystyle{\pi}}$

7.   $\displaystyle{\csc\, (\arcsin\, \frac{3}{5})}$   $\ans{\displaystyle{\frac{5}{3}}}$

8.   $\displaystyle{\textrm{arcsec}\, (\sin \frac{\pi}{2})}$   $\ans{\displaystyle{0}}$

9.   $\displaystyle{\sin(\textrm{arcsec}\, 2)}$   $\ans{\displaystyle{\frac{\sqrt{3}}{2}}}$

10.   $\displaystyle{\arccos\, (-\frac{1}{2})}$   $\ans{\displaystyle{\frac{2\pi}{3}}}$

11.   $\displaystyle{\arctan^3\, (-\sqrt{3})}$   $\ans{\displaystyle{-\frac{\pi^3}{27}}}$

12.   $\displaystyle{3\arcsin^2\, \frac{\sqrt{3}}{2}}$   $\ans{\displaystyle{\frac{\pi^2}{3}}}$

13.   $\displaystyle{\textrm{arcsec}\, 0}$   $\ans{\displaystyle{\textrm{no value}}}$

1.   $\displaystyle{\sin\, (\arctan\, 2)}$   $\ans{\displaystyle{\frac{2}{\sqrt{5}}}}$

2.   $\displaystyle{\arccos\, (\sin\, (-\frac{\pi}{6}))}$   $\ans{\displaystyle{\frac{2\pi}{3}}}$

3.   $\displaystyle{\tan\, (\arccos\, (-\frac{2}{3}))}$   $\ans{\displaystyle{-\frac{\sqrt{5}}{2}}}$

4.   $\displaystyle{\arccos\, 2}$   $\ans{\displaystyle{\textrm{no value}}}$

5.   $\displaystyle{\cos\, (\arcsin\, (-\frac{4}{5}))}$   $\ans{\displaystyle{\frac{3}{5}}}$

6.   $\displaystyle{4 \arctan\, 1}$   $\ans{\displaystyle{\pi}}$

7.   $\displaystyle{\csc\, (\textrm{arcsec}\, 12)}$   $\ans{\displaystyle{\frac{12}{\sqrt{143}}}}$

8.   $\displaystyle{\textrm{arccsc}\, \sqrt{2}}$   $\ans{\displaystyle{\frac{\pi}{4}}}$

9.   $\displaystyle{\textrm{arcsec}\, 2}$   $\ans{\displaystyle{\frac{\pi}{3}}}$

10.   $\displaystyle{\arctan\, (\sin \frac{\pi}{2})}$   $\ans{\displaystyle{\frac{\pi}{4}}}$

11.   $\displaystyle{\arctan\, (\cos\, \pi)}$   $\ans{\displaystyle{-\frac{\pi}{4}}}$

12.   $\displaystyle{\arcsin\, (\tan \frac{\pi}{4})}$   $\ans{\displaystyle{\frac{\pi}{2}}}$

3. Write the given expression in terms of $x$ without any trigonometric functions.

1.   $\displaystyle{\sin\, (\arctan\, x)}$   $\ans{\displaystyle{\frac{x}{\sqrt{x^2+1}}}}$

2.   $\displaystyle{\tan\, (\arcsin\, x)}$   $\ans{\displaystyle{\frac{x}{\sqrt{1-x^2}}}}$

3.   $\displaystyle{\cot\, (\arcsin\, x)}$   $\ans{\displaystyle{\frac{\sqrt{1-x^2}}{x}}}$

1.   $\displaystyle{\cos\, (\arcsin\, x)}$   $\ans{\displaystyle{\sqrt{1-x^2}}}$

2.   $\displaystyle{\cos\, (\textrm{arcsec}\, x)}$   $\ans{\displaystyle{\frac{1}{x}}}$

3.   $\displaystyle{\csc\, (\textrm{arccot}\, \frac{x}{4})}$   $\ans{\displaystyle{\frac{\sqrt{x^2+16}}{4}}}$