91Ó°ÊÓ

Inverse trigonometric functions are essential when you need to find angles based on their trigonometric ratios.
In our problem, we use the inverse sine function, denoted as \(\text{sin}^{-1} \(x\)\).
This function gives us the angle \(\theta\) whose sine is x.
For example, \(\text{sin}^{-1} \frac{3}{5}\) means we need to find an angle \(\theta\) such that \(\text{sin}(\theta) = \frac{3}{5}\).
A key thing to remember is that the range of \(\text{sin}^{-1} x\) is between \(-\frac{\text{Ï€}}{2}\) and \(\frac{\text{Ï€}}{2}\), ensuring we get a unique angle.

The Pythagorean identity is a fundamental equation in trigonometry: \(\text{sin}^2(\theta) + \text{cos}^2(\theta) = 1\).
This relationship allows us to find one trigonometric ratio if we know the other.
In our problem, after finding \(\text{sin}(\theta) = \frac{3}{5}\), we use the identity to discover \(\text{cos}(\theta)\).
We substitute and solve: \(\frac{9}{25} + \text{cos}^2(\theta) = 1\).
Rearranging and solving gives \(\text{cos}^2(\theta) = \frac{16}{25}\), leading to two possible values: \(\text{cos}(\theta)= \frac{4}{5}\) or \(-\frac{4}{5}\).
Since our angle \(\theta\) is between \(-\frac{\text{Ï€}}{2}\) and \(\frac{\text{Ï€}}{2}\), we choose the positive value \(\frac{4}{5}\).

The double angle formula for cosine is useful to simplify expressions featuring twice an angle: \(\text{cos}(2\theta) = 2 \text{cos}^2(\theta) - 1\).
Given \(\text{cos}(\theta) = \frac{4}{5}\), we can find the cosine of twice the angle, \(\text{cos}(2\theta)\).
Substituting in our value, we get: \(\text{cos}(2\theta) = 2(\frac{4}{5})^2 - 1\).
Simplifying gives: \(\text{cos}(2\theta) = 2(\frac{16}{25}) - 1 = \frac{32}{25} - \frac{25}{25} = \frac{7}{25}\).
Thus, the exact value for \(\text{cos}(2 \text{sin}^{-1} \frac{3}{5})\) is \(\frac{7}{25}\).