Problem 90 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify values of a and b

Given expression is \(\cos 120^{\circ}+\cos 60^{\circ}\). In the sum-to-product formula \(2 \cos a \cos b = \cos(a+b) + \cos(a-b)\), let \(a = \frac{(120+60)}{2} = 90^{\circ}\) and \(b = \frac{(120-60)}{2} = 30^{\circ}\)

Step 2: Apply Sum-to-product formula

By substituting \(a = 90^{\circ}\), \(b = 30^{\circ}\) into sum-product formula we get, \(\cos 120^{\circ}+\cos 60^{\circ} = 2 \cos 90^{\circ} \cos 30^{\circ}\)

Step 3: Calculate the value

We know that, \(\cos 90^{\circ} = 0\) and \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\). So, substituting these values we have, \(2 \cos 90^{\circ} \cos 30^{\circ} = 2 * 0 * \frac{\sqrt{3}}{2} = 0\)