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Chapter 5: Problem 13

Find the exact value of each expression. $$\sin \left(45^{\circ}-30^{\circ}\right)$$

Short Answer

Expert verified
The answer to \(\sin \left(45^{\circ}-30^{\circ}\right)\) is \(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\)

Step by step solution

01

Recall the Sine Difference Identities

We should remember the following formula: \(\sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta\)
02

Substitute the Values into the Equation

We are given the following equation, \(\sin \left(45^{\circ}-30^{\circ}\right)\). Now, we will substitute the values into the Equation: \(\sin(45 - 30) = \sin(45) \cos(30) - \cos(45) \sin(30)\)
03

Calculate the given trigonometric values

We know that \(\sin(45)\) = \(\cos(45)\) = \(\frac{\sqrt{2}}{2}\), \(\sin(30)\) = \(\frac{1}{2}\) and \(\cos(30)\) = \(\frac{\sqrt{3}}{2}\). After putting these values in to the equation, we will obtain: \(\frac{\sqrt{2}}{2} * \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} * \frac{1}{2}\)
04

Calculate the final result

The final calculation leads to: \(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\)

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Most popular questions from this chapter

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.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The most efficient way that I can simplify \(\frac{(\sec x+1)(\sec x-1)}{\sin ^{2} x}\) is to immediately rewrite the expression in terms of cosines and sines.

Use the appropriate values from Exercise 110 to answer each of the following. a. Is \(\sin \left(2 \cdot 30^{\circ}\right),\) or \(\sin 60^{\circ},\) equal to \(2 \sin 30^{\circ} ?\) b. Is \(\sin \left(2 \cdot 30^{\circ}\right),\) or \(\sin 60^{\circ},\) equal to \(2 \sin 30^{\circ} \cos 30^{\circ} ?\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The word identity is used in different ways in additive identity, multiplicative identity, and trigonometric identity.

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. $$\frac{\sin x}{1-\cos ^{2} x}=\csc x$$
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