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