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Chapter 6: Problem 86

Use words to describe the formula for: the power-reducing formula for the cosine squared of an angle.

Short Answer

Expert verified
The power-reducing formula for the cosine squared of an angle is given by \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \). This formula means that the square of the cosine of any angle is equal to the average of 1 and the cosine of twice that angle.

Step by step solution

01

Identify the original formula

The power-reducing formula for the cosine squared of an angle is \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \). This formula represents a relationship between the square of the cosine of an angle and the cosine of twice that angle.
02

Break down the left-hand side of the formula

The term \( \cos^2(x) \) is the square of the cosine of an angle. Squaring a number (or function) is equivalent to multiplying that number (or function) by itself. Here, \( \cos^2(x) \) means 'the cosine of x multiplied by the cosine of x'.
03

Break down the right-hand side of the formula

On the right-hand side of the equation, \( \cos(2x) \) represents the cosine of an angle that is twice as large as x. The '+' operator adds this to 1, and the entire sum is divided by 2 which averages the value.
04

Interpret the entire formula

The formula \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \) shows that the square of the cosine of an angle can be calculated as the average of 1 and the cosine of twice that angle. This transformation can simplify computations in various contexts, particularly when integrating certain trigonometric functions.

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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. $$ \cos x-5=3 \cos x+6 $$

Write each trigonometric expression as an algebraic expression (that is, without any trigonometric functions). Assume that x and y are positive and in the domain of the given inverse trigonometric function. $$ \tan \left(\sin ^{-1} x+\cos ^{-1} y\right) $$

A ball on a spring is pulled 4 inches below its rest position and then released. After I seconds the ball's distance, \(d,\) in inches from its rest position is given by $$ d=-4 \cos \frac{\pi}{3} t $$ Find all values of \(t\) for which the ball is 2 inches below its rest position.

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Determine whether each statement makes sense or does not make sense, and explain your reasoning. After the difference formula for cosines is verified, I noticed that the other sum and difference formulas are verified relatively quickly.
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