Solved on Mar 05, 2024

Solve $\cot x(\tan x-\sqrt{3})=0$ on $[0,2\pi)$ . Select: A. $x=0,\pi/3,2\pi/3,\pi,4\pi/3,5\pi/3$ , or B. There is no solution.

STEP 1

Assumptions
1. We are solving the equation $\cot x(\tan x-\sqrt{3})=0$ .
2. The interval for the solution is $[0, 2\pi)$ .
3. We will use the properties of trigonometric functions to find the solutions.

STEP 2

The given equation can be split into two separate equations using the zero product property. If the product of two factors is zero, at least one of the factors must be zero.
$\cot x = 0 \quad \text{or} \quad \tan x - \sqrt{3} = 0$

STEP 3

Solve the first equation $\cot x = 0$ . Recall that $\cot x = \frac{1}{\tan x}$ , so this equation implies that $\tan x$ is undefined, which occurs when $x$ is an odd multiple of $\frac{\pi}{2}$ within the given interval.

STEP 4

Find the values of $x$ within the interval $[0, 2\pi)$ where $\tan x$ is undefined.
$x = \frac{\pi}{2}, \frac{3\pi}{2}$

STEP 5

Solve the second equation $\tan x - \sqrt{3} = 0$ . This implies that $\tan x = \sqrt{3}$ .

STEP 6

Recall that $\tan x = \sqrt{3}$ corresponds to an angle of $\frac{\pi}{3}$ in the standard position, where the tangent function is positive. This occurs in the first and third quadrants.

STEP 7

Find the values of $x$ within the interval $[0, 2\pi)$ where $\tan x = \sqrt{3}$ .
$x = \frac{\pi}{3}, \frac{4\pi}{3}$

STEP 8

Combine the solutions from steps 4 and 7 to get all the solutions within the interval $[0, 2\pi)$ .
$x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{3}, \frac{4\pi}{3}$

STEP 9

Write the final solution, ensuring that it is in the correct format as requested in the problem statement.
A. $x = \frac{\pi}{3}, \frac{\pi}{2}, \frac{4\pi}{3}, \frac{3\pi}{2}$

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Solve $\cot x(\tan x-\sqrt{3})=0$ on $[0,2\pi)$ . Select: A. $x=0,\pi/3,2\pi/3,\pi,4\pi/3,5\pi/3$ , or B. There is no solution.

STEP 1

Assumptions
1. We are solving the equation $\cot x(\tan x-\sqrt{3})=0$ .
2. The interval for the solution is $[0, 2\pi)$ .
3. We will use the properties of trigonometric functions to find the solutions.

STEP 2

The given equation can be split into two separate equations using the zero product property. If the product of two factors is zero, at least one of the factors must be zero.
$\cot x = 0 \quad \text{or} \quad \tan x - \sqrt{3} = 0$

STEP 3

Solve the first equation $\cot x = 0$ . Recall that $\cot x = \frac{1}{\tan x}$ , so this equation implies that $\tan x$ is undefined, which occurs when $x$ is an odd multiple of $\frac{\pi}{2}$ within the given interval.

STEP 4

Find the values of $x$ within the interval $[0, 2\pi)$ where $\tan x$ is undefined.
$x = \frac{\pi}{2}, \frac{3\pi}{2}$

STEP 5

Solve the second equation $\tan x - \sqrt{3} = 0$ . This implies that $\tan x = \sqrt{3}$ .

STEP 6

Recall that $\tan x = \sqrt{3}$ corresponds to an angle of $\frac{\pi}{3}$ in the standard position, where the tangent function is positive. This occurs in the first and third quadrants.

STEP 7

Find the values of $x$ within the interval $[0, 2\pi)$ where $\tan x = \sqrt{3}$ .
$x = \frac{\pi}{3}, \frac{4\pi}{3}$

STEP 8

Combine the solutions from steps 4 and 7 to get all the solutions within the interval $[0, 2\pi)$ .
$x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{3}, \frac{4\pi}{3}$

STEP 9

Write the final solution, ensuring that it is in the correct format as requested in the problem statement.
A. $x = \frac{\pi}{3}, \frac{\pi}{2}, \frac{4\pi}{3}, \frac{3\pi}{2}$

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Solve cot⁡x(tan⁡x−3)=0\cot x(\tan x-\sqrt{3})=0cotx(tanx−3​)=0 on [0,2π)[0,2\pi)[0,2π). Select: A. x=0,π/3,2π/3,π,4π/3,5π/3x=0,\pi/3,2\pi/3,\pi,4\pi/3,5\pi/3x=0,π/3,2π/3,π,4π/3,5π/3, or B. There is no solution.

STEP 1

Assumptions1. We are solving the equation cot⁡x(tan⁡x−3)=0\cot x(\tan x-\sqrt{3})=0cotx(tanx−3​)=0.2. The interval for the solution is [0,2π)[0, 2\pi)[0,2π).3. We will use the properties of trigonometric functions to find the solutions.

STEP 2

The given equation can be split into two separate equations using the zero product property. If the product of two factors is zero, at least one of the factors must be zero.cot⁡x=0ortan⁡x−3=0\cot x = 0 \quad \text{or} \quad \tan x - \sqrt{3} = 0cotx=0ortanx−3​=0

STEP 3

Solve the first equation cot⁡x=0\cot x = 0cotx=0. Recall that cot⁡x=1tan⁡x\cot x = \frac{1}{\tan x}cotx=tanx1​, so this equation implies that tan⁡x\tan xtanx is undefined, which occurs when xxx is an odd multiple of π2\frac{\pi}{2}2π​ within the given interval.

STEP 4

Find the values of xxx within the interval [0,2π)[0, 2\pi)[0,2π) where tan⁡x\tan xtanx is undefined.x=π2,3π2x = \frac{\pi}{2}, \frac{3\pi}{2}x=2π​,23π​

STEP 5

Solve the second equation tan⁡x−3=0\tan x - \sqrt{3} = 0tanx−3​=0. This implies that tan⁡x=3\tan x = \sqrt{3}tanx=3​.

STEP 6

Recall that tan⁡x=3\tan x = \sqrt{3}tanx=3​ corresponds to an angle of π3\frac{\pi}{3}3π​ in the standard position, where the tangent function is positive. This occurs in the first and third quadrants.

STEP 7

Find the values of xxx within the interval [0,2π)[0, 2\pi)[0,2π) where tan⁡x=3\tan x = \sqrt{3}tanx=3​.x=π3,4π3x = \frac{\pi}{3}, \frac{4\pi}{3}x=3π​,34π​

STEP 8

Combine the solutions from steps 4 and 7 to get all the solutions within the interval [0,2π)[0, 2\pi)[0,2π).x=π2,3π2,π3,4π3x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{3}, \frac{4\pi}{3}x=2π​,23π​,3π​,34π​

STEP 9

Write the final solution, ensuring that it is in the correct format as requested in the problem statement.A. x=π3,π2,4π3,3π2x = \frac{\pi}{3}, \frac{\pi}{2}, \frac{4\pi}{3}, \frac{3\pi}{2}x=3π​,2π​,34π​,23π​

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Solve $\cot x(\tan x-\sqrt{3})=0$ on $[0,2\pi)$ . Select: A. $x=0,\pi/3,2\pi/3,\pi,4\pi/3,5\pi/3$ , or B. There is no solution.

Assumptions
1. We are solving the equation $\cot x(\tan x-\sqrt{3})=0$ .
2. The interval for the solution is $[0, 2\pi)$ .
3. We will use the properties of trigonometric functions to find the solutions.

The given equation can be split into two separate equations using the zero product property. If the product of two factors is zero, at least one of the factors must be zero.
$\cot x = 0 \quad \text{or} \quad \tan x - \sqrt{3} = 0$

Solve the first equation $\cot x = 0$ . Recall that $\cot x = \frac{1}{\tan x}$ , so this equation implies that $\tan x$ is undefined, which occurs when $x$ is an odd multiple of $\frac{\pi}{2}$ within the given interval.

Find the values of $x$ within the interval $[0, 2\pi)$ where $\tan x$ is undefined.
$x = \frac{\pi}{2}, \frac{3\pi}{2}$

Solve the second equation $\tan x - \sqrt{3} = 0$ . This implies that $\tan x = \sqrt{3}$ .

Recall that $\tan x = \sqrt{3}$ corresponds to an angle of $\frac{\pi}{3}$ in the standard position, where the tangent function is positive. This occurs in the first and third quadrants.

Find the values of $x$ within the interval $[0, 2\pi)$ where $\tan x = \sqrt{3}$ .
$x = \frac{\pi}{3}, \frac{4\pi}{3}$

Combine the solutions from steps 4 and 7 to get all the solutions within the interval $[0, 2\pi)$ .
$x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{3}, \frac{4\pi}{3}$

Write the final solution, ensuring that it is in the correct format as requested in the problem statement.
A. $x = \frac{\pi}{3}, \frac{\pi}{2}, \frac{4\pi}{3}, \frac{3\pi}{2}$