Solved on Nov 08, 2023

Given $\cos 2x = \frac{4}{5}$ with $\frac{\pi}{2} < x < \pi$ , find $\cos x = \pm \sqrt{\frac{3}{5}}$ .

STEP 1

Assumptions1. The given value is \cosx = \frac{4}{5} . The range for $x$ is $\frac{\pi}{}<x<\pi$
3. We need to find the exact value of $\cos x$

STEP 2

We can use the double-angle identity for cosine, which states that $\cos2x =2\cos^2 x -1$ . We can rearrange this equation to solve for $\cos x$ .
$\cos x = \sqrt{\frac{\cos2x +1}{2}}$

STEP 3

Substitute the given value of $\cos2x$ into the equation.
$\cos x = \sqrt{\frac{/5 +1}{2}}$

STEP 4

implify the fraction inside the square root.
$\cos x = \sqrt{\frac{9/}{2}}$

STEP 5

Continue simplifying the fraction inside the square root.
$\cos x = \sqrt{\frac{9}{10}}$

STEP 6

Take the square root to find the value of $\cos x$ .
$\cos x = \frac{3}{\sqrt{10}}$

STEP 7

Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{10}$ .
$\cos x = \frac{3\sqrt{10}}{10}$ However, we know that the range for $x$ is $\frac{\pi}{2}<x<\pi$ , which means that $\cos x$ should be negative (since cosine is negative in the second quadrant). Therefore, we need to adjust our answer to reflect this.

STEP 8

Adjust the sign of $\cos x$ to reflect the correct quadrant.
$\cos x = -\frac{3\sqrt{10}}{10}$ So, the exact value of $\cos x$ is $-\frac{3\sqrt{10}}{10}$ .

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Given $\cos 2x = \frac{4}{5}$ with $\frac{\pi}{2} < x < \pi$ , find $\cos x = \pm \sqrt{\frac{3}{5}}$ .

STEP 1

Assumptions1. The given value is \cosx = \frac{4}{5} . The range for $x$ is $\frac{\pi}{}<x<\pi$
3. We need to find the exact value of $\cos x$

STEP 2

We can use the double-angle identity for cosine, which states that $\cos2x =2\cos^2 x -1$ . We can rearrange this equation to solve for $\cos x$ .
$\cos x = \sqrt{\frac{\cos2x +1}{2}}$

STEP 3

Substitute the given value of $\cos2x$ into the equation.
$\cos x = \sqrt{\frac{/5 +1}{2}}$

STEP 4

implify the fraction inside the square root.
$\cos x = \sqrt{\frac{9/}{2}}$

STEP 5

Continue simplifying the fraction inside the square root.
$\cos x = \sqrt{\frac{9}{10}}$

STEP 6

Take the square root to find the value of $\cos x$ .
$\cos x = \frac{3}{\sqrt{10}}$

STEP 7

STEP 8

Adjust the sign of $\cos x$ to reflect the correct quadrant.
$\cos x = -\frac{3\sqrt{10}}{10}$ So, the exact value of $\cos x$ is $-\frac{3\sqrt{10}}{10}$ .

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Given cos⁡2x=45\cos 2x = \frac{4}{5}cos2x=54​ with π2<x<π\frac{\pi}{2} < x < \pi2π​<x<π, find cos⁡x=±35\cos x = \pm \sqrt{\frac{3}{5}}cosx=±53​​.

STEP 1

Assumptions1. The given value is \cosx = \frac{4}{5} . The range for xxx is π<x<π\frac{\pi}{}<x<\piπ​<x<π3. We need to find the exact value of cos⁡x\cos xcosx

STEP 2

We can use the double-angle identity for cosine, which states that cos⁡2x=2cos⁡2x−1\cos2x =2\cos^2 x -1cos2x=2cos2x−1. We can rearrange this equation to solve for cos⁡x\cos xcosx.cos⁡x=cos⁡2x+12\cos x = \sqrt{\frac{\cos2x +1}{2}}cosx=2cos2x+1​​

STEP 3

Substitute the given value of cos⁡2x\cos2xcos2x into the equation.cos⁡x=/5+12\cos x = \sqrt{\frac{/5 +1}{2}}cosx=2/5+1​​

STEP 4

implify the fraction inside the square root.cos⁡x=9/2\cos x = \sqrt{\frac{9/}{2}}cosx=29/​​

STEP 5

Continue simplifying the fraction inside the square root.cos⁡x=910\cos x = \sqrt{\frac{9}{10}}cosx=109​​

STEP 6

Take the square root to find the value of cos⁡x\cos xcosx.cos⁡x=310\cos x = \frac{3}{\sqrt{10}}cosx=10​3​

STEP 7

STEP 8

Adjust the sign of cos⁡x\cos xcosx to reflect the correct quadrant.cos⁡x=−31010\cos x = -\frac{3\sqrt{10}}{10}cosx=−10310​​So, the exact value of cos⁡x\cos xcosx is −31010-\frac{3\sqrt{10}}{10}−10310​​.

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Given $\cos 2x = \frac{4}{5}$ with $\frac{\pi}{2} < x < \pi$ , find $\cos x = \pm \sqrt{\frac{3}{5}}$ .

Assumptions1. The given value is \cosx = \frac{4}{5} . The range for $x$ is $\frac{\pi}{}<x<\pi$
3. We need to find the exact value of $\cos x$

We can use the double-angle identity for cosine, which states that $\cos2x =2\cos^2 x -1$ . We can rearrange this equation to solve for $\cos x$ .
$\cos x = \sqrt{\frac{\cos2x +1}{2}}$

Substitute the given value of $\cos2x$ into the equation.
$\cos x = \sqrt{\frac{/5 +1}{2}}$

implify the fraction inside the square root.
$\cos x = \sqrt{\frac{9/}{2}}$

Continue simplifying the fraction inside the square root.
$\cos x = \sqrt{\frac{9}{10}}$

Take the square root to find the value of $\cos x$ .
$\cos x = \frac{3}{\sqrt{10}}$

Adjust the sign of $\cos x$ to reflect the correct quadrant.
$\cos x = -\frac{3\sqrt{10}}{10}$ So, the exact value of $\cos x$ is $-\frac{3\sqrt{10}}{10}$ .