# Solve $2\cos^2(x)=1$: Find $\theta$

• MHB
• karush
In summary, the conversation discusses finding the value of $x$ given the equation $2\cos^2(x)=1$. The expert summarizer provides two methods for solving the equation, one using $\theta$ instead of $x$ and the other using the double angle identity for cosine. The conversation also mentions the possibility of using the cosine addition formula to solve the equation.
karush
Gold Member
MHB
$\tiny{\textsf{9c Miliani HS hw}}$
Find x
$2\cos^2(x)=1$
$\cos^2(x)=\dfrac{1}{2}\implies \cos(x)=\dfrac{1}{\sqrt{2}}\implies x=\dfrac{\pi}{4}$ or $x=\dfrac{7\pi}{4}$

well its the simple ones where we stumple
not sure if it necessary to give all quadrants of possible answers
its probably better to us $\theta$ rather than x
mahalo

You forgot $-1 \over root2$ as the other solution.
There are two methods one $cos^2x = cos^2\alpha$ , $x= n \pi + \alpha$ and $x = n \pi - \alpha$
and second method $cosx = cos\alpha , x = 2n \pi + \alpha , x = 2n \pi - \alpha$
take any alpha which satisfies the value and you will get whole possible answers.

well that's pretty valuable to know
i was thinking of just one revolution

$2\cos^2{x} -1 = 0$

$\cos(2x) = 0$

$2x = (2k+1) \cdot \dfrac{\pi}{2} \implies x = (2k+1) \cdot \dfrac{\pi}{4} \, , \, x \in \mathbb{Z}$

ok not sure where 2x= came from but see that it works

karush said:
ok not sure where 2x= came from but see that it works

double angle identity for cosine derivation …

$\cos(2x) = \cos(x+x) = \cos^2{x} - \sin^2{x} = \cos^2{x} - (1-\cos^2{x}) = 2\cos^2{x} -1$

also, note …

$\cos(2x) = \cos^2{x}-\sin^2{x} = (1-\sin^2{x})-\sin^2{x} = 1-2\sin^2{x}$

He might also don't know about $cos(A+B) = cosA cosB - sinA sinB$ and all other formulas

yes i know that one its very common

## What does the equation $2\cos^2(x)=1$ mean?

This equation is asking for the value of the angle $\theta$ (represented by $x$) that satisfies the equation. In other words, it is asking for the value of $\theta$ that makes the equation true.

## How do I solve $2\cos^2(x)=1$?

To solve this equation, you can use algebraic techniques to manipulate the equation and isolate $\theta$ on one side. Then, you can use a calculator or a trigonometric table to find the inverse cosine of the resulting value and determine the value of $\theta$.

## What are the possible values of $\theta$ that satisfy $2\cos^2(x)=1$?

Since cosine is a periodic function, there are infinitely many values of $\theta$ that satisfy this equation. However, we typically restrict our solutions to a specific interval, such as $[0, 2\pi)$ or $[-\pi, \pi)$.

## Why does $2\cos^2(x)=1$ have two solutions?

This equation has two solutions because cosine is a periodic function and has a period of $2\pi$. This means that there are two distinct angles that have the same cosine value, and both of these angles satisfy the equation.

## What is the relationship between $2\cos^2(x)=1$ and the unit circle?

The equation $2\cos^2(x)=1$ can be rewritten as $\cos^2(x)=\frac{1}{2}$. This means that the cosine of $\theta$ is equal to $\frac{1}{2}$, which is the $x$-coordinate of a point on the unit circle. Therefore, the solutions to this equation correspond to the angles on the unit circle where the $x$-coordinate is $\frac{1}{2}$.

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