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Chapter 5: Problem 99

Use the sum-to-product formulas to find the exact value of the expression. $$\sin 75^{\circ}+\sin 15^{\circ}$$

Short Answer

Expert verified
The exact value of the expression \(\sin 75掳 + \sin 15掳 = \frac{\sqrt{6}}{2}\)

Step by step solution

01

Identify the Parts of the Identity

In our sum-to-product identity, we will let A represent the 75掳 and B represent the 15掳. So we are dealing with A = 75掳 and B = 15掳.
02

Application of Sum-to-Product Identity

Applying the sum-to-product identity formula \(\sin A + \sin B = 2 \sin \frac{{A + B}}{2} \cos \frac{{A - B}}{2}\), we substitute the values A = 75掳 and B = 15掳 into the identity to get \(2 \sin \frac{{75 + 15}}{2} \cos \frac{{75 - 15}}{2}\) which simplifies to \(2 \sin 45掳 \cos 30掳\)
03

Evaluate the Trigonometric Values

Evaluate the trigonometric values \(\sin 45掳\) and \(\cos 30掳\). By referring to standard values, we can determine that \(\sin 45掳 = \frac{\sqrt{2}}{2}\) and \(\cos 30掳 = \frac{\sqrt{3}}{2}\)
04

Solve for the Expression

Multiply the values together. The expression \(2 \sin 45掳 \cos 30掳\) becomes \(2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2}\)

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