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Chapter 5: Problem 37

Find the exact value of the expression. $$\sin \frac{\pi}{12} \cos \frac{\pi}{4}+\cos \frac{\pi}{12} \sin \frac{\pi}{4}$$

Short Answer

Expert verified
The exact value of the given expression is \(\frac{\sqrt{3}}{2}\).

Step by step solution

01

Recognise the format of the equation

On examining the expression, it can be noticed that it is using the Angle Sum Identity for sine which is \(\sin(a+b) = \sin a \cos b + \cos a \sin b\). The given expression is \(\sin \frac{\pi}{12} \cos \frac{\pi}{4}+\cos \frac{\pi}{12} \sin \frac{\pi}{4}\). Therefore, it can be rewritten as \( \sin(\frac{\pi}{12}+ \frac{\pi}{4})\).
02

Simplify the sum in the sine function

The sum inside the sine function can be simplified: \(\sin(\frac{\pi}{12} + \frac{\pi}{4}) = \sin(\frac{pi}{12} + \frac{3\pi}{12}) = \sin(\frac{4\pi}{12}). Simplifying further, we get \sin(\frac{\pi}{3}).
03

Substitute the Value

Looking up the exact value of \(\sin(\frac{\pi}{3})\) which is usually provided in the unit circle in trigonometry class, we find it is \(\frac{\sqrt{3}}{2}\). So, the original equation simplifies to \(\frac{\sqrt{3}}{2}\).

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