Practice Inverse Trigonometric Functions in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Functions that reverse the trigonometric functions: given a ratio, they return the corresponding angle. \arcsin, \arccos, and \arctan are the inverses of \sin, \cos, and \tan on restricted domains.
Regular trig functions answer: 'Given an angle, what's the ratio?' Inverse trig functions answer the reverse: 'Given a ratio, what's the angle?' Since \sin and \cos are many-to-one (many angles give the same ratio), we must restrict their domains to make the inverse a proper function. Think of it like this: if you know the slope of a ramp is 0.5, \arcsin(0.5) = 30° tells you the angle.
Example 1
easyEvaluate \arcsin\!\left(\frac{1}{2}\right), \arccos\!\left(-\frac{\sqrt{2}}{2}\right), and \arctan(1). State the range of each inverse trig function.
Example 2
hardSimplify \sin(\arccos(x)) for x\in[-1,1] without trigonometric functions in the final answer.
Example 3
easyA right triangle has opposite side 5 and hypotenuse 13. Find the angle \theta opposite the side of length 5 using \arcsin.
Example 4
mediumSolve 2\sin(x)-1=0 on [0,2\pi] using \arcsin, and explain why there are two solutions.