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Chapter 5: Problem 16

Find the exact value of each expression. $$\sin 75^{\circ}$$

Short Answer

Expert verified
The exact value of \(\sin 75^{\circ}\) is \(\frac{\sqrt{2} + \sqrt{6}}{4}\).

Step by step solution

01

Break Down the Angle

Break 75 degrees down into the sum of two angles. For example, it can be written as \(45^{\circ} + 30^{\circ}\).
02

Substitute the Values

Substitute the values into the sum of angles identity: \(\sin(45^{\circ} + 30^{\circ}) = \sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ}\).
03

Find the Sine and Cosine Values

We know that \(\sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2}\), \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\), and \(\sin 30^{\circ} = \frac{1}{2}\). Substitute these values into the equation.
04

Simplify the Expression

Simplify the expression and the final result will be \(\frac{\sqrt{2} + \sqrt{6}}{4}\).

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. To prove a trigonometric identity, I select one side of the equation and transform it until it is the other side of the equation, or I manipulate both sides to a common trigonometric expression.

Remembering the six sum and difference identities can be difficult. Did you have problems with some exercises because the identity you were using in your head turned out to be an incorrect formula? Are there easy ways to remember the six new identities presented in this section? Group members should address this question, considering one identity at a time. For each formula, list ways to make it casier to remember.

In the interval \([0,2 \pi),\) the solutions of \(\sin x=\cos 2 x\) are \(\frac{\pi}{6}, \frac{5 \pi}{6},\) and \(\frac{3 \pi}{2} .\) Explain how to use graphs generated by a graphing utility to check these solutions.

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Graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$\sin x=-\cos x \tan (-x)$$
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