91Ó°ÊÓ

To find the value of $$\cos^{-1} \frac{3}{4}$$, we need to find the angle whose cosine is $$\frac{3}{4}$$. Let's call this angle $$\theta$$ such that $$\cos \theta = \frac{3}{4}$$. Since $$\cos \theta = \frac{3}{4}$$, we can set up the relationship as $$\theta = \cos^{-1} \frac{3}{4}$$.

Now that we know the value of $$\theta$$, we can add the two angles $$\frac{\pi}{6}$$ and $$\theta$$ together as: \[ \alpha = \frac{\pi}{6} + \theta \]

With the value of $$\alpha$$, we can now evaluate the final expression: \[ \cos \left( \frac{\pi}{6} + \cos^{-1} \frac{3}{4} \right) = \cos (\alpha) \] To get the final value for $$\cos (\alpha)$$, it is important to note that we may need to resort to approximations, as an exact value may not be available. If an exact value is desired or if the angle is a recognizable value, we can use the trigonometric identity for the cosine of the sum of two angles which states that: \[ \cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) \] In our case, $$\alpha = \frac{\pi}{6}$$ and $$\beta = \cos^{-1} \frac{3}{4}$$. By substituting and evaluating the trigonometric identity, we can arrive at the final value of the given expression.