Question

Find the exact value of the given expression. $$ \sin ^{-1}\left(\frac{\sqrt{3}}{2}\right) $$

Short answer

The exact value of the given expression \(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\) is \(\frac{1}{3}\pi\).

Step-by-step solution

Determine the trigonometrical identity

We have the expression \(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\). Recall the definition of inverse sine: If \(\sin{\theta} = x\), then \(\sin^{-1}{x} = \theta\). So, we are looking for an angle \(\theta\) such that \(\sin{\theta} = \frac{\sqrt{3}}{2}\).

Identify an angle with sine equal to \(\frac{\sqrt{3}}{2}\)

We can use the properties of special right triangles and the unit circle to determine the angle. In particular, consider a 30-60-90 right triangle. Recall that the ratios of side lengths for such a triangle are 1:\(\frac{\sqrt{3}}{2}\):\(\frac{1}{2}\). So, if \(\theta = 60^{\circ}\), we have \(\sin{\left(60^{\circ}\right)} = \frac{\sqrt{3}}{2}\). Note that since sine is positive in both the first and second quadrants, we also have another angle that satisfies the condition, which is \(180^{\circ} - 60^{\circ} = 120^{\circ}\). However, since the inverse sine function is defined in the range \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), the only valid angle for this problem is \(60^{\circ}\).

Convert the angle into radians

Our final step is to convert the angle we found, \(60^{\circ}\), into radians. Remember that \(180^{\circ} = \pi\) radians. So, \(\theta = 60^{\circ} \cdot \frac{\pi}{180^{\circ}} = \frac{1}{3}\pi\) radians.

Write down the exact value of the given expression

We have found the angle \(\theta\) such that \(\sin{\theta} = \frac{\sqrt{3}}{2}\): \(\theta = \frac{1}{3}\pi\) radians. Therefore, \[\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{1}{3}\pi.\]