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Chapter 6: Problem 89

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 \(\sin 75^{\circ}+\sin 15^{\circ}\) is \(\sqrt{6} - \sqrt{2}\)

Step by step solution

01

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.
02

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: \( 2 \sin \frac{1}{2}(75^{\circ}+15^{\circ}) \cos \frac{1}{2}(75^{\circ}-15^{\circ})\)
03

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.
04

Evaluate the expression

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

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Most popular questions from this chapter

The graph of a function \(f\) is shown over the 122, the graph of a function \(f\) is shown over the interval \([\mathbf{0}, \mathbf{2} \pi] .\) (a) Find the \(x\) -intercepts of the graph of \(f\) algebraically. Verify your solutions by using the zero or root feature of a graphing utility. (b) The \(x\) -coordinates of the extrema of \(f\) are solutions of the trigonometric equation. (Calculus is required to find the trigonometric equation.) Find the solutions of the equation algebraically. Verify these solutions using the maximum and minimum features of the graphing utility. Function: \(f(x)=\cos 2 x+\sin x\) Trigonometric Equation: \(-2 \sin 2 x+\cos x=0\)

Use the sum-to-product formulas to write the sum or difference as a product. $$\sin (\alpha+\beta)-\sin (\alpha-\beta)$$

Use the product-to-sum formulas to write the product as a sum or difference. $$6 \sin 45^{\circ} \cos 15^{\circ}$$

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

Find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\sin u=\frac{5}{13}, \quad \pi / 2
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