Inverse Trigonometric Functions

Inverse Trigonometric Functions - Examples on ITF

Example 11: Find the value of $\arcsin\left(-\frac{1}{2}\right)$
- Solution: Since $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$, we have $\arcsin\left(-\frac{1}{2}\right) = -\frac{\pi}{6}$
Example 12: Solve for x: $\sin^{-1}(x) = -\frac{\pi}{6}$
- Solution: Using the definition of inverse sine function, we get $x = \sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2}$
Example 13: Calculate the value of $\arccos\left(-\frac{\sqrt{3}}{2}\right)$
- Solution: Since $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$, we have $\arccos\left(-\frac{\sqrt{3}}{2}\right) = \frac{5\pi}{6}$
Example 14: Solve for x: $\cos^{-1}(x) = \frac{\pi}{4}$
- Solution: Using the definition of inverse cosine function, we get $x = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
Example 15: Find the value of $\arctan\left(-\sqrt{3}\right)$
- Solution: Since $\tan\left(-\frac{\pi}{3}\right) = -\sqrt{3}$, we have $\arctan\left(-\sqrt{3}\right) = -\frac{\pi}{3}$

Example 16: Solve for x: $\tan^{-1}(x) = -\frac{\pi}{4}$
- Solution: Using the definition of inverse tangent function, we get $x = \tan\left(-\frac{\pi}{4}\right) = -1$
Example 17: Calculate the value of $\arccsc\left(1\right)$
- Solution: Since $\csc\left(\frac{\pi}{2}\right) = 1$, we have $\arccsc\left(1\right) = \frac{\pi}{2}$
Example 18: Solve for x: $\csc^{-1}(x) = \frac{\pi}{6}$
- Solution: Using the definition of inverse cosecant function, we get $x = \csc\left(\frac{\pi}{6}\right) = 2$
Example 19: Find the value of $\arcsec\left(2\right)$
- Solution: Since $\sec\left(\frac{\pi}{3}\right) = 2$, we have $\arcsec\left(2\right) = \frac{\pi}{3}$
Example 20: Solve for x: $\sec^{-1}(x) = \frac{\pi}{4}$
- Solution: Using the definition of inverse secant function, we get $x = \sec\left(\frac{\pi}{4}\right) = \sqrt{2}$

Example 21: Find the value of $\arcsin\left(\frac{\sqrt{2}}{2}\right)$
- Solution: Since $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$, we have $\arcsin\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}$
Example 22: Solve for x: $\sin^{-1}(x) = \frac{\pi}{3}$
- Solution: Using the definition of inverse sine function, we get $x = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$
Example 23: Calculate the value of $\arccos\left(\frac{1}{2}\right)$
- Solution: Since $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$, we have $\arccos\left(\frac{1}{2}\right) = \frac{\pi}{3}$
Example 24: Solve for x: $\cos^{-1}(x) = \frac{\pi}{6}$
- Solution: Using the definition of inverse cosine function, we get $x = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$
Example 25: Find the value of $\arctan\left(1\right)$
- Solution: Since $\tan\left(\frac{\pi}{4}\right) = 1$, we have $\arctan\left(1\right) = \frac{\pi}{4}$

Example 26: Solve for x: $\tan^{-1}(x) = \pi$
- Solution: Using the definition of inverse tangent function, we get $x = \tan(\pi) = 0$
Example 27: Calculate the value of $\arccsc\left(\sqrt{2}\right)$
- Solution: Since $\csc\left(\frac{\pi}{4}\right) = \sqrt{2}$, we have $\arccsc\left(\sqrt{2}\right) = \frac{\pi}{4}$
Example 28: Solve for x: $\csc^{-1}(x) = \frac{2\pi}{3}$
- Solution: Using the definition of inverse cosecant function, we get $x = \csc\left(\frac{2\pi}{3}\right) = -\frac{2\sqrt{3}}{3}$
Example 29: Find the value of $\arcsec\left(-1\right)$
- Solution: Since $\sec\left(\pi\right) = -1$, we have $\arcsec\left(-1\right) = \pi$
Example 30: Solve for x: $\sec^{-1}(x) = -\frac{\pi}{2}$
- Solution: Using the definition of inverse secant function, we get $x = \sec\left(-\frac{\pi}{2}\right) = \text{undefined}$