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Chapter 5: Problem 58

Use the sum-to-product formulas to find the exact value of the expression. $$\cos 120^{\circ}+\cos 60^{\circ}$$

Short Answer

Expert verified
The exact value of the expression \(\cos 120^{\circ}+\cos 60^{\circ}\) is 0.

Step by step solution

01

Identify A and B

The given expression is \(\cos 120^{\circ}+\cos 60^{\circ}\). In this, 120 degrees is 'A' and 60 degrees is 'B'.
02

Apply Sum-to-Product Formula

The sum-to-product formula for cosine is \(\cos A + \cos B = 2 \cos\left(\frac{{A+B}}{2}\right)\cos\left(\frac{{A-B}}{2}\right)\). Using this formula, the given expression becomes \(2 \cos\left(\frac{{120^{\circ} + 60^{\circ}}}{2}\right)\cos\left(\frac{{120^{\circ} - 60^{\circ}}}{2}\right)\) or \(2 \cos 90^{\circ}\cos 30^{\circ}\) .
03

Calculate the Cosine Values

The cosine of 90 degrees is 0 and the cosine of 30 degrees is \(\frac{\sqrt{3}}{2}\). Substituting these values gives \[2 \cdot 0 \cdot \frac{\sqrt{3}}{2} = 0\].
04

Give the Final Answer

From step 3, the value of the expression \(\cos 120^{\circ}+\cos 60^{\circ}\) is found to be zero.

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