Math / Trigonometry

QuestionWhat is the value of $\tan \left(\frac{2 \pi}{3}+x\right)$ if $\tan x=2 \sqrt{3}$ ? Step $3 \Rightarrow(2 \sin x \cos x) \frac{\sin x}{\cos x}$ Step $4 \Rightarrow 2 \sin ^{2} x$ (B) $\frac{\sqrt{3}}{7}$ (C) $\frac{3 \sqrt{3}}{7}$ (D) $\frac{-3 \sqrt{3}}{7}$

Studdy Solution

STEP 1

What is this asking? If the tangent of an angle $x$ is $2\sqrt{3}$ , what's the tangent of $\frac{2\pi}{3} + x$ ? Watch out! Don't forget your tangent addition formula, and remember to simplify completely!

STEP 2

1. Tangent Addition Formula
2. Substitute and Simplify

STEP 3

Alright, let's kick things off with the tangent addition formula!
It tells us how to find the tangent of the sum of two angles.
Specifically, $\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a) \cdot \tan(b)}$ .

STEP 4

In our case, $a$ is $\frac{2\pi}{3}$ and $b$ is $x$ , so we've got $\tan\left(\frac{2\pi}{3} + x\right) = \frac{\tan\left(\frac{2\pi}{3}\right) + \tan(x)}{1 - \tan\left(\frac{2\pi}{3}\right) \cdot \tan(x)}$ .
This formula is super important, so make sure you have it memorized!

STEP 5

We know that $\tan(x) = 2\sqrt{3}$ , and we can find $\tan\left(\frac{2\pi}{3}\right)$ using the unit circle, which gives us $-\sqrt{3}$ .
Let's plug these values in: $\tan\left(\frac{2\pi}{3} + x\right) = \frac{-\sqrt{3} + 2\sqrt{3}}{1 - (-\sqrt{3}) \cdot 2\sqrt{3}}$ .

STEP 6

Now, let's simplify the numerator: $-\sqrt{3} + 2\sqrt{3} = \sqrt{3}$ .
In the denominator, we have $1 - (-\sqrt{3} \cdot 2\sqrt{3}) = 1 - (-6) = 1 + 6 = 7$ .

STEP 7

Putting it all together, we get $\tan\left(\frac{2\pi}{3} + x\right) = \frac{\sqrt{3}}{7}$ .
Boom!

STEP 8

The value of $\tan\left(\frac{2\pi}{3} + x\right)$ is $\frac{\sqrt{3}}{7}$ .

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QuestionWhat is the value of $\tan \left(\frac{2 \pi}{3}+x\right)$ if $\tan x=2 \sqrt{3}$ ? Step $3 \Rightarrow(2 \sin x \cos x) \frac{\sin x}{\cos x}$ Step $4 \Rightarrow 2 \sin ^{2} x$ (B) $\frac{\sqrt{3}}{7}$ (C) $\frac{3 \sqrt{3}}{7}$ (D) $\frac{-3 \sqrt{3}}{7}$

Studdy Solution

STEP 1

What is this asking? If the tangent of an angle $x$ is $2\sqrt{3}$ , what's the tangent of $\frac{2\pi}{3} + x$ ? Watch out! Don't forget your tangent addition formula, and remember to simplify completely!

STEP 2

1. Tangent Addition Formula
2. Substitute and Simplify

STEP 3

Alright, let's kick things off with the tangent addition formula!
It tells us how to find the tangent of the sum of two angles.
Specifically, $\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a) \cdot \tan(b)}$ .

STEP 4

In our case, $a$ is $\frac{2\pi}{3}$ and $b$ is $x$ , so we've got $\tan\left(\frac{2\pi}{3} + x\right) = \frac{\tan\left(\frac{2\pi}{3}\right) + \tan(x)}{1 - \tan\left(\frac{2\pi}{3}\right) \cdot \tan(x)}$ .
This formula is super important, so make sure you have it memorized!

STEP 5

We know that $\tan(x) = 2\sqrt{3}$ , and we can find $\tan\left(\frac{2\pi}{3}\right)$ using the unit circle, which gives us $-\sqrt{3}$ .
Let's plug these values in: $\tan\left(\frac{2\pi}{3} + x\right) = \frac{-\sqrt{3} + 2\sqrt{3}}{1 - (-\sqrt{3}) \cdot 2\sqrt{3}}$ .

STEP 6

Now, let's simplify the numerator: $-\sqrt{3} + 2\sqrt{3} = \sqrt{3}$ .
In the denominator, we have $1 - (-\sqrt{3} \cdot 2\sqrt{3}) = 1 - (-6) = 1 + 6 = 7$ .

STEP 7

Putting it all together, we get $\tan\left(\frac{2\pi}{3} + x\right) = \frac{\sqrt{3}}{7}$ .
Boom!

STEP 8

The value of $\tan\left(\frac{2\pi}{3} + x\right)$ is $\frac{\sqrt{3}}{7}$ .

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Studdy Solution

STEP 1

What is this asking? If the tangent of an angle xxx is 232\sqrt{3}23​, what's the tangent of 2π3+x\frac{2\pi}{3} + x32π​+x? Watch out! Don't forget your tangent addition formula, and remember to simplify completely!

STEP 2

1. Tangent Addition Formula2. Substitute and Simplify

STEP 3

STEP 4

STEP 5

STEP 6

Now, let's **simplify** the numerator: −3+23=3-\sqrt{3} + 2\sqrt{3} = \sqrt{3}−3​+23​=3​.In the denominator, we have 1−(−3⋅23)=1−(−6)=1+6=71 - (-\sqrt{3} \cdot 2\sqrt{3}) = 1 - (-6) = 1 + 6 = 71−(−3​⋅23​)=1−(−6)=1+6=7.

STEP 7

Putting it all together, we get tan⁡(2π3+x)=37\tan\left(\frac{2\pi}{3} + x\right) = \frac{\sqrt{3}}{7}tan(32π​+x)=73​​.Boom!

STEP 8

The value of tan⁡(2π3+x)\tan\left(\frac{2\pi}{3} + x\right)tan(32π​+x) is 37\frac{\sqrt{3}}{7}73​​.

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Studdy solves anything!

Start learning now

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

What is this asking? If the tangent of an angle $x$ is $2\sqrt{3}$ , what's the tangent of $\frac{2\pi}{3} + x$ ? Watch out! Don't forget your tangent addition formula, and remember to simplify completely!

1. Tangent Addition Formula
2. Substitute and Simplify

Now, let's simplify the numerator: $-\sqrt{3} + 2\sqrt{3} = \sqrt{3}$ .
In the denominator, we have $1 - (-\sqrt{3} \cdot 2\sqrt{3}) = 1 - (-6) = 1 + 6 = 7$ .

Putting it all together, we get $\tan\left(\frac{2\pi}{3} + x\right) = \frac{\sqrt{3}}{7}$ .
Boom!

The value of $\tan\left(\frac{2\pi}{3} + x\right)$ is $\frac{\sqrt{3}}{7}$ .