Solve the equation for all solutions in the interval [0, 27).10) /2 cos 2x = 1

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Solve the equation for all solutions in the interval [0, 27t). 10) V2 Cos 2x = 1 %3D

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.4: Multiple-angle Formulas

Problem 44E

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

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A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

Solve the equation for all solutions in the interval [0, 27).
10) /2 cos 2x = 1

Expert Solution

Concept Used

Given: $\sqrt{2} \cos (2 x) = 1$

To find all x values in the interval $[0, 2 π)$

The cosine function is positive in the 1st and 4th quadrants.

Step 2

To find the general solution let isolate the cosine function and the value of $\cos (\frac{π}{4}) = \frac{1}{\sqrt{2}}$ cosine function is positive in first and fourth quadrant using $\cos (2 π - x) = \cos (x)$

$\begin{array}{rcl} \sqrt{2} \cos (2 x) & = & 1 \\ \cos (2 x) & = & \frac{1}{\sqrt{2}} \\ \cos (2 x) & = & \cos (\frac{π}{4}) \\ 2 x & = & \frac{π}{4} \\ x & = & \frac{π}{8} \\ \cos (2 π - \frac{π}{4}) & = & \cos (\frac{π}{4}) \\ \cos (\frac{7 π}{4}) & = & \cos (2 x) \\ \frac{7 π}{4} & = & 2 x \\ \frac{7 π}{8} & = & x \\ 2 x & = & \pm 2 n π + \frac{π}{4} and \pm 2 n π + \frac{7 π}{4} \\ x & = & \pm \frac{2 nπ}{2} + \frac{π}{4 \cdot 2} and \pm \frac{2 nπ}{2} + \frac{7 π}{4 \cdot 2} \\ x & = & \pm n π + \frac{π}{8} and \pm n π + \frac{7 π}{8} \end{array}$

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