Math / Trigonometry

Question $4 \cos 2 x=3$

Studdy Solution

STEP 1

1. The equation $4 \cos 2x = 3$ is a trigonometric equation.
2. We will need to isolate the cosine term and use inverse trigonometric functions to solve for $x$ .

STEP 2

1. Isolate the cosine term $\cos 2x$ .
2. Solve for $2x$ using the inverse cosine function.
3. Solve for $x$ .

STEP 3

First, divide both sides of the equation by 4 to isolate the cosine term:
$4 \cos 2x = 3$ $\cos 2x = \frac{3}{4}$

STEP 4

Use the inverse cosine function to solve for $2x$ . The general solution for $\cos \theta = a$ is $\theta = \cos^{-1}(a) + 2k\pi$ and $\theta = -\cos^{-1}(a) + 2k\pi$ , where $k$ is an integer:
$2x = \cos^{-1}\left(\frac{3}{4}\right) + 2k\pi$ $2x = -\cos^{-1}\left(\frac{3}{4}\right) + 2k\pi$

STEP 5

Solve for $x$ by dividing each part of the equation by 2:
$x = \frac{1}{2} \cos^{-1}\left(\frac{3}{4}\right) + k\pi$ $x = -\frac{1}{2} \cos^{-1}\left(\frac{3}{4}\right) + k\pi$
The general solutions for $x$ are:
$x = \frac{1}{2} \cos^{-1}\left(\frac{3}{4}\right) + k\pi$ $x = -\frac{1}{2} \cos^{-1}\left(\frac{3}{4}\right) + k\pi$
where $k$ is an integer.

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