/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 67 Find a formula for \(\sin (4 \th... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 67

Find a formula for \(\sin (4 \theta)\) in terms of \(\cos \theta\) and \(\sin \theta\).

Short Answer

Expert verified
The formula for \(\sin(4\theta)\) in terms of \(\sin\theta\) and \(\cos\theta\) is: \(\sin(4\theta) = 4 \sin\theta \cos\theta (\cos^2\theta - \sin^2\theta)\)

Step by step solution

01

Write down the double-angle formulas for sine and cosine

Recall the double-angle formulas for sine and cosine: 1. \(\sin(2\alpha) = 2 \sin\alpha \cos\alpha\) 2. \(\cos(2\beta) = \cos^2\beta - \sin^2\beta\) We will be using these formulas in the next steps.
02

Apply the double-angle formula for sine to obtain an expression for sin(4胃)

Now, we can replace the angle 伪 with 2胃 in the double-angle formula for sine: \(\sin(2(2\theta)) = \sin(4\theta)\) Using the formula \(\sin(2\alpha) = 2 \sin\alpha \cos\alpha\): \(\sin(4\theta) = 2 \sin(2\theta) \cos(2\theta)\) Next, we need expressions for \(\sin(2\theta)\) and \(\cos(2\theta)\) using the double-angle formulas and the angle 胃.
03

Find expressions for sin(2胃) and cos(2胃) using the double-angle formulas

Recalling the double-angle formulas, we replace 伪 with 胃 and 尾 with 胃: 1. \(\sin(2\theta) = 2 \sin\theta \cos\theta\) 2. \(\cos(2\theta) = \cos^2\theta - \sin^2\theta\) Now we have expressions for \(\sin(2\theta)\) and \(\cos(2\theta)\) that only involve \(\sin\theta\) and \(\cos\theta\).
04

Substitute the expressions for sin(2胃) and cos(2胃) back into the expression for sin(4胃)

Substitute the expressions for \(\sin(2\theta)\) and \(\cos(2\theta)\) from Step 3 into the expression for \(\sin(4\theta)\): \(\sin(4\theta) = 2 (2 \sin\theta \cos\theta)(\cos^2\theta - \sin^2\theta)\)
05

Simplify the expression for sin(4胃)

Now, simplify the expression by distributing the factor 2: \(\sin(4\theta) = 4 \sin\theta \cos\theta (\cos^2\theta - \sin^2\theta)\) This is the formula for \(\sin(4\theta)\) in terms of the sine and cosine of 胃.

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Most popular questions from this chapter

Show that $$\cos (2 \theta) \leq \cos ^{2} \theta$$ for every angle \(\theta\).

Find exact expressions for the indicated quantities. The following information will be useful: $$ \begin{array}{l} \cos 22.5^{\circ}=\frac{\sqrt{2+\sqrt{2}}}{2} \text { and } \sin 22.5^{\circ}=\frac{\sqrt{2-\sqrt{2}}}{2} \\ \cos 18^{\circ}=\sqrt{\frac{\sqrt{5}+5}{8}} \text { and } \sin 18^{\circ}=\frac{\sqrt{5}-1}{4} \end{array} $$ [The value for \(\sin 22.5^{\circ}\) used here was derived in Example 4 in Section \(5.5 ;\) the other values were derived in Exercise 64 and Problems 102 and 103 in Section \(5.5 .]\) $$\sin 37.5^{\circ}$$

Show that $$(\cos \theta+\sin \theta)^{2}(\cos \theta-\sin \theta)^{2}+\sin ^{2}(2 \theta)=1$$ for all angles \(\theta\).

Evaluate the indicated expressions assuming that $$ \begin{array}{ll} \cos x=\frac{1}{3} & \text { and } \sin y=\frac{1}{4} \\ \sin u=\frac{2}{3} & \text { and } \cos v=\frac{1}{5} \end{array} $$ Assume also that \(x\) and \(u\) are in the interval \(\left(0, \frac{\pi}{2}\right),\) that \(y\) is in the interval \(\left(\frac{\pi}{2}, \pi\right),\) and that \(v\) is in the interval \(\left(-\frac{\pi}{2}, 0\right) .\) $$\cos (u-v)$$

Show that $$\tan \frac{x}{4}=\frac{\sqrt{2-2 \cos x}-\sin x}{1-\cos x}$$ for all \(x\) in the interval \((0,2 \pi)\). [Hint: Start with a half-angle formula for tangent to express \(\tan \frac{x}{4}\) in terms of \(\sin \frac{x}{2}\) and \(\cos \frac{x}{2}\). Then use half-angle formulas for cosine and sine, along with algebraic manipulations.]
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