Problem 89 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The exact value of the expression \(\sin 75^{\circ}+\sin 15^{\circ}\) is \(\sqrt{6} - \sqrt{2}\)

Step 1: Identify the sum-to-product formula

The appropriate sum-to-product formula in this case is: \( \sin A + \sin B = 2 \sin \frac{1}{2}(A+B) \cos \frac{1}{2}(A-B) \). This formula will be used to rewrite the expression into a format that can be evaluated.

Step 2: Substitute the given values

We will use the given values and plug them into the formula. Hence, \( A = 75^{\circ} \) and \( B = 15^{\circ} \). Substituting these into the formula gives: \( 2 \sin \frac{1}{2}(75^{\circ}+15^{\circ}) \cos \frac{1}{2}(75^{\circ}-15^{\circ})\)

Step 3: Simplify the expression

Now simplify the expression to: \(2 \sin 45^{\circ} \cos 30^{\circ}\). The values for \(\sin 45^{\circ}\) and \(\cos 30^{\circ}\) are well known.

Step 4: Evaluate the expression

Evaluate the expression: \(2 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} = \sqrt{6} - \sqrt{2}\)