Problem 91 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the sum-to-product formulas to find the exact value of the expression. $$\cos \frac{3 \pi}{4}-\cos \frac{\pi}{4}$$

Step-by-Step Solution

Answer

-\sqrt{2}

Step 1: Recognize the form

The provided expression corresponds to \( -2 \sin \frac{a+b}{2} \sin \frac{a-b}{2} \) where \( a = \frac{3\pi}{4} \) and \( b = \frac{\pi}{4} \)

Step 2: Plug the values into the formula

On using these values in the formula, we get \( -2 \sin \left( \frac{\frac{3\pi}{4} + \frac{\pi}{4}}{2}\right) \sin \left( \frac{\frac{3\pi}{4} - \frac{\pi}{4}}{2}\right) \)

Step 3: Simplify the expression

Simplifying the expression, we get \( -2 \sin \left( \frac{\pi}{2}\right) \sin \left( \frac{\pi}{4}\right) \)

Step 4: Calculate the sine values

Let's calculate the sine values of these two angles. \( \sin \frac{\pi}{2} = 1 \) and \( \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \)

Step 5: Complete the Calculation

Plugging these sine values back into the expression, we get \( -2 \times 1 \times \frac{1}{\sqrt{2}} = -\sqrt{2} \)