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Trigonometric Identities and Equations
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Find all solutions of the equation in the interval
[0,2pi).

cot ((2theta)/(3))+sqrt3=0
Write your answer in radians in terms of
pi. If there is more than one solution, separate them with commas.

theta=

◻

piquad◻,◻,dots
Aa

◻
Explanation
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Math Al $\newline$ Best Al Homework Hel $\newline$ Logging in $\newline$ A $\newline$ ALEKS - Jameson Colle $\newline$ ByteLearn $\newline$ https://Mww-awu.aleks.com/alekscgi/x/Isl.exe/ $1$ o_u-IgNslkasNW $8$ D... $\newline$ Import favorites $\newline$ MVNU Students Ho... $\newline$ A $\newline$ ALEKS - Jameson C... $\newline$ Trigonometric Identities and Equations $\newline$ Finding solutions in an interval for a trigonometric equation involving an... $\newline$ $0 / 5$ $\newline$ Jameson $\newline$ Español $\newline$ Find all solutions of the equation in the interval $[0,2 \pi)$ . $\newline$ $\cot \frac{2 \theta}{3}+\sqrt{3}=0$ $\newline$ Write your answer in radians in terms of $\pi$ . If there is more than one solution, separate them with commas. $\newline$ $\theta=$ $\newline$ $\square$ $\newline$ $\pi \quad \square, \square, \ldots$ $\newline$ Aa $\newline$ $\square$ $\newline$ Explanation $\newline$ Check $\newline$ C $2024$ McGraw Hill LLC. All Rights Reserved. $\newline$ Terms of Use $\newline$ Privacy Center $\newline$ Accessibility $\newline$ Type here to search $\newline$ $60^{\circ} \mathrm{F}$ $\newline$ ( $1$ )) $\newline$ $2$ : $49$ PM $\newline$ $4 / 22 / 2024$

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Math Problems

Grade 6

Solve one-step multiplication and division equations: word problems

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Q. Math Al

\newline

Best Al Homework Hel

\newline

Logging in

\newline

\newline

ALEKS - Jameson Colle

\newline

ByteLearn

\newline

https://Mww-awu.aleks.com/alekscgi/x/Isl.exe/

1

o_u-IgNslkasNW

8

D...

\newline

Import favorites

\newline

MVNU Students Ho...

\newline

\newline

ALEKS - Jameson C...

\newline

Trigonometric Identities and Equations

\newline

Finding solutions in an interval for a trigonometric equation involving an...

\newline

0 / 5

\newline

Jameson

\newline

Español

\newline

Find all solutions of the equation in the interval

[0,2 \pi)

\newline

\cot \frac{2 \theta}{3}+\sqrt{3}=0

\newline

Write your answer in radians in terms of

\pi

. If there is more than one solution, separate them with commas.

\newline

\theta=

\newline

\square

\newline

\pi \quad \square, \square, \ldots

\newline

\newline

\square

\newline

Explanation

\newline

Check

\newline

2024

\newline

\newline

Privacy Center

\newline

Accessibility

\newline

Type here to search

\newline

60^{\circ} \mathrm{F}

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(

1

))

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2

49

\newline

4 / 22 / 2024

Rewrite Equation: Rewrite the equation $\cot\left(\frac{2\theta}{3}\right) + \sqrt{3} = 0$ as $\cot\left(\frac{2\theta}{3}\right) = -\sqrt{3}.$
Identify Tangent Value: Recognize that $\cot(x) = -\sqrt{3}$ corresponds to an angle where the tangent has a value of $-\frac{1}{\sqrt{3}}$ , which is $-\frac{\pi}{6}$ or $\frac{11\pi}{6}$ in the unit circle.
Find Corresponding Angles: Since we have $\cot\left(\frac{2\theta}{3}\right)$ , we need to find the values of $\frac{2\theta}{3}$ that correspond to $-\frac{\pi}{6}$ and $\frac{11\pi}{6}$ .
Solve for Theta: Set $(2\theta)/3$ equal to $-\pi/6$ and solve for theta: $(2\theta)/3 = -\pi/6$ , so $2\theta = -\pi/2$ , which gives $\theta = -\pi/4$ . But this is not in the interval $[0, 2\pi)$ , so we discard it.
Consider Equivalent Angles: Set $(2\theta)/3$ equal to $11\pi/6$ and solve for $\theta$ : $(2\theta)/3 = 11\pi/6$ , so $2\theta = 11\pi/2$ , which gives $\theta = 11\pi/4$ . This is also not in the interval $[0, 2\pi)$ , so we need to find equivalent angles within the interval.
Subtract Multiples of $2\pi$ : To find equivalent angles for $\theta$ within $[0, 2\pi)$ , we can subtract multiples of $2\pi$ from $\frac{11\pi}{4}$ until the angle is within the interval. Subtracting $2\pi$ ( $\frac{8\pi}{4}$ ) from $\frac{11\pi}{4}$ gives $\frac{3\pi}{4}$ , which is in the interval.
Add Period for More Solutions: We also need to consider that cotangent has a period of $\pi$ , so we can add $\pi$ to $\frac{3\pi}{4}$ to find another solution within the interval. Adding $\pi$ ( $\frac{4\pi}{4}$ ) to $\frac{3\pi}{4}$ gives $\frac{7\pi}{4}$ , which is also in the interval.
Check for Additional Solutions: Check for any other solutions by adding or subtracting multiples of $\pi$ to our found solutions, $\frac{3\pi}{4}$ and $\frac{7\pi}{4}$ , to ensure we have all solutions within the interval $[0, 2\pi)$ . Adding $\pi$ to $\frac{7\pi}{4}$ would exceed $2\pi$ , so no further solutions are found.