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Chapter 6: Problem 92

Without showing algebraic details, describe in words how to reduce the power of \(\cos ^{4} x\).

Short Answer

Expert verified
\(\cos^{4}x = \frac{3}{8} + \frac{1}{2} \cos(2x) + \frac{1}{8} \cos(4x)\)

Step by step solution

01

Understand the power-reduction formula

The power-reducing formula for \(\cos^{2}x\) is \(\cos^{2}x = \frac{1+ \cos(2x)}{2}\). This is an identity in trigonometry used for reducing the power of 2 of a cosine function into a simple equation involving the cosine of twice the angle.
02

Apply the power-reduction formula to \(\cos^{4}x\)

To express \(\cos^{4}x\) in terms of \(\cos(2x)\) and \(\cos(4x)\), start by understanding that \(\cos^{4}x = (\cos^{2}x)^2\). Then, substitute the power-reducing formula of \(\cos^{2}x\) into this equation, which gives \((\frac{1+ \cos(2x)}{2})^2\). Then, square this to finally get \(\frac{1}{4} + \frac{1}{2} \cos(2x) + \frac{1}{4} \cos^2(2x)\).
03

Apply the power-reduction formula again

Now, the term with \(\cos^2(2x)\) needs to be reduced further. Applying the power-reduction formula to this term yields \(\cos^{4}x = \frac{1}{4} + \frac{1}{2} \cos(2x) + \frac{1}{4}( \frac{1+ \cos(4x)}{2} )\). The final expression for \(\cos^{4}x\) in terms of \(\cos(2x)\) and \(\cos(4x)\) is \(\cos^{4}x = \frac{3}{8} + \frac{1}{2} \cos(2x) + \frac{1}{8} \cos(4x)\).

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Most popular questions from this chapter

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$ 5 \sin x=2 \cos ^{2} x-4 $$

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 that the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$ \cos \left(\frac{3 \pi}{2}-x\right)=-\sin x $$

Solve each equation on the interval \([0,2 \pi)\) Do not use a calculator. $$ \sin 3 x+\sin x+\cos x=0 $$

Exercises \(110-112\) will help you prepare for the material covered in the next section. 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 \(2 \sin 30^{\circ} \cos 30^{\circ} ?\)

Find the exact value of each expression. Do not use a calculator. $$ \cos \left(\tan ^{-1} \frac{4}{3}+\cos ^{-1} \frac{5}{13}\right) $$
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