Questions: Use a sum or difference formula to find the exact value of the trigonometric function. cos(5π/12) The exact value of cos(5π/12) is □. (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Solution Steps

To find the exact value of $\cos \frac{5\pi}{12}$, we can use the sum or difference formula for cosine. We can express $\frac{5\pi}{12}$ as a sum or difference of angles whose cosine values are known. One possible way is to write $\frac{5\pi}{12}$ as $\frac{\pi}{4} + \frac{\pi}{6}$. Then, apply the cosine sum formula: $\cos(a + b) = \cos a \cos b - \sin a \sin b$.

To find $\cos \frac{5\pi}{12}$, we can express $\frac{5\pi}{12}$ as a sum of two angles: $$\frac{5\pi}{12} = \frac{\pi}{4} + \frac{\pi}{6}$$

Using the cosine sum formula, we have: $$\cos(a + b) = \cos a \cos b - \sin a \sin b$$ where $a = \frac{\pi}{4}$ and $b = \frac{\pi}{6}$.

We know the following values: $$\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \quad \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$$ $$\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}, \quad \sin \frac{\pi}{6} = \frac{1}{2}$$

Substituting these values into the formula gives: $$\cos \frac{5\pi}{12} = \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right)$$

Calculating the terms: $$\cos \frac{5\pi}{12} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}$$

Final Answer