Problem 35
Question
Find the exact value of the expression. $$\sin \frac{\pi}{12} \cos \frac{\pi}{4}+\cos \frac{\pi}{12} \sin \frac{\pi}{4}$$
Step-by-Step Solution
Verified Answer
\(\sqrt{3}/2\)
1Step 1: Recognize and apply the sine addition formula
The given expression is in the form of the sine addition formula where \(A = \pi/12\) and \(B = \pi/4\). Applying the formula we get \(\sin (\pi/12 + \pi/4)\).
2Step 2: Simplify the addition
Simplify the addition in the argument of the sine function. Express all angles in terms of \(\pi\). \(\pi/12 + \pi/4 = \pi/12 + 3\pi/12 = 4\pi/12 = \pi/3\). Thus we get \(\sin (\pi/3)\).
3Step 3: Substitute the exact value
From the unit circle, we know that the exact value of \(\sin (\pi/3)\) is \(\sqrt{3}/2\).
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