Solved on Mar 28, 2024

Solve the equation $2 \cos x - \sqrt{3} = 0$ and provide a general formula for all solutions.

STEP 1

1. The equation $2 \cos x - \sqrt{3} = 0$ involves the cosine function, which is periodic with period $2\pi$ .
2. The solutions to the equation will be in the form $x = \theta + 2k\pi$ or $x = -\theta + 2k\pi$ , where $\theta$ is a specific solution in the range $[0, 2\pi)$ and $k$ is an integer.
3. The value of $\cos x$ is $\sqrt{3}/2$ when $x$ equals certain specific angles within one period of the cosine function.

STEP 2

1. Isolate $\cos x$ in the equation.
2. Find the specific solutions for $\cos x = \sqrt{3}/2$ within one period of the cosine function.
3. Write the general formula for all solutions using the periodicity of the cosine function.

STEP 3

Isolate $\cos x$ by adding $\sqrt{3}$ to both sides of the equation.
$2 \cos x = \sqrt{3}$

STEP 4

Divide both sides of the equation by 2 to solve for $\cos x$ .
$\cos x = \frac{\sqrt{3}}{2}$

STEP 5

Identify the specific angles for which $\cos x = \frac{\sqrt{3}}{2}$ within one period $[0, 2\pi)$ .
The cosine function equals $\frac{\sqrt{3}}{2}$ at $x = \frac{\pi}{6}$ and $x = \frac{11\pi}{6}$ .

STEP 6

Write down the specific solutions within the interval $[0, 2\pi)$ .
$x = \frac{\pi}{6} \quad \text{or} \quad x = \frac{11\pi}{6}$

STEP 7

Express the general formula for all solutions using the periodicity of the cosine function.
For the solution $x = \frac{\pi}{6}$ :
$x = \frac{\pi}{6} + 2k\pi \quad \text{where} \quad k \in \mathbb{Z}$

STEP 8

Express the general formula for the second solution $x = \frac{11\pi}{6}$ :
$x = \frac{11\pi}{6} + 2k\pi \quad \text{where} \quad k \in \mathbb{Z}$
The general formula for all the solutions to the equation $2 \cos x - \sqrt{3} = 0$ is:
$x = \frac{\pi}{6} + 2k\pi \quad \text{or} \quad x = \frac{11\pi}{6} + 2k\pi \quad \text{where} \quad k \in \mathbb{Z}$

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Solve the equation $2 \cos x - \sqrt{3} = 0$ and provide a general formula for all solutions.

STEP 1

STEP 2

1. Isolate $\cos x$ in the equation.
2. Find the specific solutions for $\cos x = \sqrt{3}/2$ within one period of the cosine function.
3. Write the general formula for all solutions using the periodicity of the cosine function.

STEP 3

Isolate $\cos x$ by adding $\sqrt{3}$ to both sides of the equation.
$2 \cos x = \sqrt{3}$

STEP 4

Divide both sides of the equation by 2 to solve for $\cos x$ .
$\cos x = \frac{\sqrt{3}}{2}$

STEP 5

Identify the specific angles for which $\cos x = \frac{\sqrt{3}}{2}$ within one period $[0, 2\pi)$ .
The cosine function equals $\frac{\sqrt{3}}{2}$ at $x = \frac{\pi}{6}$ and $x = \frac{11\pi}{6}$ .

STEP 6

Write down the specific solutions within the interval $[0, 2\pi)$ .
$x = \frac{\pi}{6} \quad \text{or} \quad x = \frac{11\pi}{6}$

STEP 7

Express the general formula for all solutions using the periodicity of the cosine function.
For the solution $x = \frac{\pi}{6}$ :
$x = \frac{\pi}{6} + 2k\pi \quad \text{where} \quad k \in \mathbb{Z}$

STEP 8

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Solve the equation 2cos⁡x−3=02 \cos x - \sqrt{3} = 02cosx−3​=0 and provide a general formula for all solutions.

STEP 1

STEP 2

1. Isolate cos⁡x\cos xcosx in the equation.2. Find the specific solutions for cos⁡x=3/2\cos x = \sqrt{3}/2cosx=3​/2 within one period of the cosine function.3. Write the general formula for all solutions using the periodicity of the cosine function.

STEP 3

Isolate cos⁡x\cos xcosx by adding 3\sqrt{3}3​ to both sides of the equation.2cos⁡x=3 2 \cos x = \sqrt{3} 2cosx=3​

STEP 4

Divide both sides of the equation by 2 to solve for cos⁡x\cos xcosx.cos⁡x=32 \cos x = \frac{\sqrt{3}}{2} cosx=23​​

STEP 5

Identify the specific angles for which cos⁡x=32\cos x = \frac{\sqrt{3}}{2}cosx=23​​ within one period [0,2π)[0, 2\pi)[0,2π).The cosine function equals 32\frac{\sqrt{3}}{2}23​​ at x=π6x = \frac{\pi}{6}x=6π​ and x=11π6x = \frac{11\pi}{6}x=611π​.

STEP 6

Write down the specific solutions within the interval [0,2π)[0, 2\pi)[0,2π).x=π6orx=11π6 x = \frac{\pi}{6} \quad \text{or} \quad x = \frac{11\pi}{6} x=6π​orx=611π​

STEP 7

Express the general formula for all solutions using the periodicity of the cosine function.For the solution x=π6x = \frac{\pi}{6}x=6π​:x=π6+2kπwherek∈Z x = \frac{\pi}{6} + 2k\pi \quad \text{where} \quad k \in \mathbb{Z} x=6π​+2kπwherek∈Z

STEP 8

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Solve the equation $2 \cos x - \sqrt{3} = 0$ and provide a general formula for all solutions.

1. Isolate $\cos x$ in the equation.
2. Find the specific solutions for $\cos x = \sqrt{3}/2$ within one period of the cosine function.
3. Write the general formula for all solutions using the periodicity of the cosine function.

Isolate $\cos x$ by adding $\sqrt{3}$ to both sides of the equation.
$2 \cos x = \sqrt{3}$

Divide both sides of the equation by 2 to solve for $\cos x$ .
$\cos x = \frac{\sqrt{3}}{2}$

Identify the specific angles for which $\cos x = \frac{\sqrt{3}}{2}$ within one period $[0, 2\pi)$ .
The cosine function equals $\frac{\sqrt{3}}{2}$ at $x = \frac{\pi}{6}$ and $x = \frac{11\pi}{6}$ .

Write down the specific solutions within the interval $[0, 2\pi)$ .
$x = \frac{\pi}{6} \quad \text{or} \quad x = \frac{11\pi}{6}$

Express the general formula for all solutions using the periodicity of the cosine function.
For the solution $x = \frac{\pi}{6}$ :
$x = \frac{\pi}{6} + 2k\pi \quad \text{where} \quad k \in \mathbb{Z}$