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Chapter 5: Problem 40

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 \(\cos 120^{\circ} \cos 30^{\circ}+\sin 120^{\circ} \sin 30^{\circ}\) is 鈥�0.

Step by step solution

01

Identify the Form of the Problem

Looking at the original expression given, \(\cos 120^{\circ} \cos 30^{\circ}+\sin 120^{\circ} \sin 30^{\circ}\), this can actually be identified as resembling the formula for the cosine of sum of two angles. The cosine sum identity is: \(\cos(A+B)= \cos A \cos B -\sin A \sin B\). However, our formula has a plus instead of a minus, which may resemble more like the cosine difference formula of \(\cos(A-B)= \cos A \cos B +\sin A \sin B\). In our case A could be 120 degrees and B could be 30 degrees.
02

Apply the Formula

Since we want to simplify our expression using trigonometric identities as much as possible, observing our original formula, we can see that the expression is equivalent to \(\cos (120^{\circ}- 30^{\circ})\), coming from the idea that this is equivalent to the cosine difference formula where A = 120 and B = 30.
03

Calculate the Result

After applying the formula, we can simplify our expression as \(\cos 90^{\circ}\) since 120掳 - 30掳 = 90掳. The cosine of 90 degrees is known to be 0.

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