Problem 92 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the sum-to-product formulas to find the exact value of the expression. $$\sin \frac{5 \pi}{4}-\sin \frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

The exact value of the expression is \(-\sqrt{2}\)

Step 1: Write down the given expression

Given expression is \(\sin \dfrac{5 \pi}{4} - \sin \dfrac{3 \pi}{4}\)

Step 2: Apply the sum-to-product formula

Using the sum-to-product formula for sine, \(\sin a - \sin b = 2 \cos \left( \dfrac{a+b}{2} \right) \sin \left( \dfrac{a-b}{2} \right)\). Substituting \( a = \dfrac{5 \pi}{4} \) and \( b = \dfrac{3 \pi}{4} \), we get \(2 \cos \left( \dfrac{\dfrac{5 \pi}{4} + \dfrac{3 \pi}{4}}{2} \right) \sin \left( \dfrac{\dfrac{5 \pi}{4} - \dfrac{3 \pi}{4}}{2} \right)\)

Step 3: Simplify the expression

On simplifying the expression obtained in step 2, we get \(2 \cos \left( \pi \right) \sin \left( \dfrac{\pi}{4} \right)\). Substituting the trigonometric values, we get \(2*(-1)*\dfrac{1}{\sqrt{2}} = -\sqrt{2}\)