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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 31 (page 465)

In Problems 11鈥�34, solve each equation on the interval $0 鈮� 胃鈮� 2 蟺$
$\cos (2 胃 - \frac{蟺}{2}) = - 1$

Short Answer

Expert verified

The solution set is $\{\frac{3 蟺}{4}, \frac{7 蟺}{4}\} .$

Step by step solution

01

Step 1. Given Information

In the given problem we have to solve each equation on the interval $0 鈮� 胃鈮� 2 蟺$

$\cos (2 胃 - \frac{蟺}{2}) = - 1$

02

Step 2. In the interval [0,2π), the cosine function equals -1 at π

So, we know that $2 胃 - \frac{蟺}{2}$ must equal $蟺$ .

To find these solutions, write the general formula that gives all the solutions.

$2 胃 - \frac{蟺}{2} = 蟺$

Add $\frac{蟺}{2}$ on both side

$2 胃 - \frac{蟺}{2} = 蟺 + 2 蟺苍 2 胃 - \frac{蟺}{2} + \frac{蟺}{2} = 蟺 + 2 蟺苍 + \frac{蟺}{2} 2 胃 = 蟺路 \frac{2}{2} + 2 蟺苍路 \frac{2}{2} + \frac{蟺}{2} 2 胃 = \frac{2 蟺 + 4 蟺苍 + 蟺}{2} 2 胃 = \frac{3 蟺 + 4 蟺苍}{2} \frac{2}{2} 胃 = \frac{3 蟺 + 4 蟺苍}{2} 路 \frac{1}{2} 胃 = \frac{3 蟺 + 4 蟺苍}{4}$

03

Step 3. The general formula is θ=3π+4πn4

So the value of given function in interval $[0, 2 蟺)$ is

$胃 = \frac{3 蟺 + 4 蟺脳 0}{4} 胃 = \frac{3 蟺 + 4 蟺脳 1}{4} 胃 = \frac{3 蟺}{4} 胃 = \frac{3 蟺 + 4 蟺}{4} 胃 = \frac{3 蟺}{4} 胃 = \frac{7 蟺}{4}$

So the solution set is $\{\frac{3 蟺}{4}, \frac{7 蟺}{4}\}$ .

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Most popular questions from this chapter

If $\tan 胃 = a \tan \frac{胃}{3}$ , express $\tan \frac{胃}{3}$ in terms of a.

If $x = 2 \tan 胃$ , express $\sin (2 胃)$ as a function ofx.

Establish each identity.

$\frac{\cos^{2} 胃 - \sin^{2} 胃}{1 - \tan^{2} 胃} = \cos^{2} 胃$

find the exact value of the expression

sin 75 $掳$ + sin 15 $掳$

Establish each identity.

${(2 a \sin 胃 \cos 胃)}^{2} + a^{2} {(\cos^{2} 胃 - \sin^{2} 胃)}^{2} = a^{2}$

See all solutions

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