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Chapter 6: Problem 88

Use words to describe the formula for: the cosine of half an angle.

Short Answer

Expert verified
The cosine of half an angle is found by adding the cosine of the angle itself to one, dividing the result by two, and then taking the square root of the result. The value can be either positive or negative depending on the quadrant of the half-angle.

Step by step solution

01

Cosine Half-Angle Identity

In trigonometry, the cosine of half an angle can be found by using the half-angle identity. The cosine half-angle identity is an equation that expresses \(\cos(\frac{胃}{2})\) (the cosine of half an angle) in terms of \(\cos(胃)\) (the cosine of the angle itself).
02

Half-Angle Identity for Cosine Formula

The cosine half-angle identity is given by the formula \(\cos(\frac{胃}{2}) = 卤\sqrt{\frac{1 + \cos(胃)}{2}}\)
03

Description of the Formula in words

To find the cosine of half an angle, you add the cosine of the angle itself to one, divide the result by two, and then take the square root of the result. The value can be either positive or negative depending on the quadrant in which the half-angle lies.

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Most popular questions from this chapter

Graph \(f\) and \(g\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi .\) Then solve a trigonometric equation to determine points of intersection and identify these points on your graphs. $$ f(x)=\cos 2 x, g(x)=1-\sin x $$

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. $$ \sin 1.2 x \cos 0.8 x+\cos 1.2 x \sin 0.8 x=\sin 2 x $$

Use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$ \tan x=-5 $$

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

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. $$ 2 \sin 3 x+\sqrt{3}=0 $$
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