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Chapter 5: Problem 38

Find the exact value of the expression. $$\cos 120^{\circ} \cos 30^{\circ}+\sin 120^{\circ} \sin 30^{\circ}$$

Short Answer

Expert verified
The exact value of the expression is 0

Step by step solution

01

Find the value for each trigonometric function

The following identities are known: \(\cos 120^{\circ} = -\frac{1}{2}\), \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\), \(\sin 120^{\circ} = \frac{\sqrt{3}}{2}\), \(\sin 30^{\circ} = \frac{1}{2}\).
02

Substitute these values into the original expression

Replacing the trigonometric functions in the original expression will yield \(-\frac{1}{2} * \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} * \frac{1}{2}\). This simplifies to \(-\frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4}\)
03

Simplify the expression

-\frac{\sqrt{3}}{4} and \frac{\sqrt{3}}{4} cancel each other out, resulting in a final answer of 0

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