To solve simple trigonometric equations using inverse trigonometric functions, it’s important to know how they work. The inverse trigonometric functions—like arcsine ($\sin^{-1}$), arccosine ($\cos^{-1}$), and arctangent ($\tan^{-1}$)—help us find angles when we have certain trigonometric ratios.

Steps to Solve

1. Identify the Equation: Start with a simple trigonometric equation like $\sin(x) = k$, where $k$ is a number between $-1$ and $1$. Not all angles will solve this equation, so it's important to use the right inverse function.

2. Apply Inverse Functions: Use the inverse function to find the angle:

   • For $\sin(x) = k$, you can find $x$ by using $x = \sin^{-1}(k)$.
   • For $\cos(x) = k$, use $x = \cos^{-1}(k)$.
   • For $\tan(x) = k$, take $x = \tan^{-1}(k)$.

3. Consider the Range: The outcomes of inverse functions are limited:

   • $\sin^{-1}(k)$ gives results between $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
   • $\cos^{-1}(k)$ gives results between $[0, \pi]$.
   • $\tan^{-1}(k)$ has outcomes in $(-\frac{\pi}{2}, \frac{\pi}{2})$.

4. Find All Solutions: Trigonometric functions repeat their values. This means you need to look for more solutions based on the original equation’s range. For sine and cosine, you can add $2\pi n$ or $(-1)^n$ to find other solutions, where $n$ is any whole number.

5. Check Your Answers: Plug your answers back into the original equations to make sure they work.

By following these steps, you can easily solve trigonometric equations. This will help you understand the relationships between angles in trigonometry better.