Find All Complex Solutions tan(theta)=(2 square root of 3)/3sin(theta)

$\tan (θ) = \frac{2 \sqrt{3}}{3} \sin (θ)$

Step 1

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $3$ .

$\tan (θ) \cdot \frac{3}{3} = \frac{2 \sqrt{3}}{3} \cdot \sin (θ) \cdot \frac{3}{3}$

Step 2

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $3$ .

$\tan (θ) \cdot 3$

Step 3

Move $3$ to the left of $\tan (θ)$ .

$\frac{3 \tan (θ)}{3} = \frac{2 \sqrt{3}}{3} \cdot \sin (θ) \cdot \frac{3}{3}$

Step 4

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $3$ .

$\sin (θ) \cdot 3$

Step 5

Move $3$ to the left of $\sin (θ)$ .

$\frac{3 \tan (θ)}{3} = \frac{2 \sqrt{3}}{3} \cdot \frac{3 \sin (θ)}{3}$

Step 6

Simplify the left side.

Simplify $\tan (θ) \cdot \frac{3}{3}$ .

Divide $3$ by $3$ .

$\tan (θ) \cdot 1 = \frac{2 \sqrt{3}}{3} \cdot \sin (θ) \cdot \frac{3}{3}$

Multiply $\tan (θ)$ by $1$ .

$\tan (θ) = \frac{2 \sqrt{3}}{3} \cdot \sin (θ) \cdot \frac{3}{3}$

Step 7

Simplify the right side.

Simplify $\frac{2 \sqrt{3}}{3} \cdot \sin (θ) \cdot \frac{3}{3}$ .

Combine $\frac{2 \sqrt{3}}{3}$ and $\sin (θ)$ .

$\tan (θ) = \frac{2 \sqrt{3} \sin (θ)}{3} \cdot \frac{3}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\tan (θ) = \frac{2 \sqrt{3} \sin (θ)}{3} \cdot \frac{3}{3}$

Rewrite the expression.

$\tan (θ) = \frac{2 \sqrt{3} \sin (θ)}{3} \cdot 1$

Multiply $\frac{2 \sqrt{3} \sin (θ)}{3}$ by $1$ .

$\tan (θ) = \frac{2 \sqrt{3} \sin (θ)}{3}$

Step 8

Divide each term in $\tan (θ) = \frac{2 \sqrt{3} \sin (θ)}{3}$ by $\tan (θ)$ and simplify.

Divide each term in $\tan (θ) = \frac{2 \sqrt{3} \sin (θ)}{3}$ by $\tan (θ)$ .

$\frac{\tan (θ)}{\tan (θ)} = \frac{\frac{2 \sqrt{3} \sin (θ)}{3}}{\tan (θ)}$

Simplify the left side.

Cancel the common factor of $\tan (θ)$ .

Cancel the common factor.

$\frac{\tan (θ)}{\tan (θ)} = \frac{\frac{2 \sqrt{3} \sin (θ)}{3}}{\tan (θ)}$

Rewrite the expression.

$1 = \frac{\frac{2 \sqrt{3} \sin (θ)}{3}}{\tan (θ)}$

Simplify the right side.

Separate fractions.

$1 = \frac{\frac{2 \sqrt{3} \sin (θ)}{3}}{1} \cdot \frac{1}{\tan (θ)}$

Rewrite $\tan (θ)$ in terms of sines and cosines.

$1 = \frac{\frac{2 \sqrt{3} \sin (θ)}{3}}{1} \cdot \frac{1}{\frac{\sin (θ)}{\cos (θ)}}$

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (θ)}{\cos (θ)}$ .

$1 = \frac{\frac{2 \sqrt{3} \sin (θ)}{3}}{1} (1 \frac{\cos (θ)}{\sin (θ)})$

Write $1$ as a fraction with denominator $1$ .

$1 = \frac{\frac{2 \sqrt{3} \sin (θ)}{3}}{1} (\frac{1}{1} \cdot \frac{\cos (θ)}{\sin (θ)})$

Simplify.

Rewrite the expression.

$1 = \frac{\frac{2 \sqrt{3} \sin (θ)}{3}}{1} (1 \frac{\cos (θ)}{\sin (θ)})$

Multiply $\frac{\cos (θ)}{\sin (θ)}$ by $1$ .

$1 = \frac{\frac{2 \sqrt{3} \sin (θ)}{3}}{1} \cdot \frac{\cos (θ)}{\sin (θ)}$

Divide $\frac{2 \sqrt{3} \sin (θ)}{3}$ by $1$ .

$1 = \frac{2 \sqrt{3} \sin (θ)}{3} \cdot \frac{\cos (θ)}{\sin (θ)}$

Cancel the common factor of $\sin (θ)$ .

Factor $\sin (θ)$ out of $2 \sqrt{3} \sin (θ)$ .

$1 = \frac{\sin (θ) (2 \sqrt{3})}{3} \cdot \frac{\cos (θ)}{\sin (θ)}$

Cancel the common factor.

$1 = \frac{\sin (θ) (2 \sqrt{3})}{3} \cdot \frac{\cos (θ)}{\sin (θ)}$

Rewrite the expression.

$1 = \frac{2 \sqrt{3}}{3} \cos (θ)$

Combine $\frac{2 \sqrt{3}}{3}$ and $\cos (θ)$ .

$1 = \frac{2 \sqrt{3} \cos (θ)}{3}$

Step 9

Rewrite the equation as $\frac{2 \sqrt{3} \cos (θ)}{3} = 1$ .

$\frac{2 \sqrt{3} \cos (θ)}{3} = 1$

Step 10

Multiply both sides of the equation by $\frac{3}{2 \sqrt{3}}$ .

$\frac{3}{2 \sqrt{3}} \cdot \frac{2 \sqrt{3} \cos (θ)}{3} = \frac{3}{2 \sqrt{3}} \cdot 1$

Step 11

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{3}{2 \sqrt{3}} \cdot \frac{2 \sqrt{3} \cos (θ)}{3}$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3}{2 \sqrt{3}} \cdot \frac{2 \sqrt{3} \cos (θ)}{3} = \frac{3}{2 \sqrt{3}} \cdot 1$

Rewrite the expression.

$\frac{1}{2 \sqrt{3}} (2 \sqrt{3} \cos (θ)) = \frac{3}{2 \sqrt{3}} \cdot 1$

Cancel the common factor of $2 \sqrt{3}$ .

Factor $2 \sqrt{3}$ out of $2 \sqrt{3} \cos (θ)$ .

$\frac{1}{2 \sqrt{3}} (2 \sqrt{3} \cos (θ)) = \frac{3}{2 \sqrt{3}} \cdot 1$

Cancel the common factor.

$\frac{1}{2 \sqrt{3}} (2 \sqrt{3} \cos (θ)) = \frac{3}{2 \sqrt{3}} \cdot 1$

Rewrite the expression.

$\cos (θ) = \frac{3}{2 \sqrt{3}} \cdot 1$

Simplify the right side.

Simplify $\frac{3}{2 \sqrt{3}} \cdot 1$ .

Multiply $\frac{3}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\cos (θ) = \frac{3}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \cdot 1$

Combine and simplify the denominator.

Multiply $\frac{3}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\cos (θ) = \frac{3 \sqrt{3}}{2 \sqrt{3} \sqrt{3}} \cdot 1$

Move $\sqrt{3}$ .

$\cos (θ) = \frac{3 \sqrt{3}}{2 (\sqrt{3} \sqrt{3})} \cdot 1$

Raise $\sqrt{3}$ to the power of $1$ .

$\cos (θ) = \frac{3 \sqrt{3}}{2 ({\sqrt{3}}^{1} \sqrt{3})} \cdot 1$

Raise $\sqrt{3}$ to the power of $1$ .

$\cos (θ) = \frac{3 \sqrt{3}}{2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})} \cdot 1$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos (θ) = \frac{3 \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}} \cdot 1$

Add $1$ and $1$ .

$\cos (θ) = \frac{3 \sqrt{3}}{2 {\sqrt{3}}^{2}} \cdot 1$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\cos (θ) = \frac{3 \sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}} \cdot 1$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cos (θ) = \frac{3 \sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}} \cdot 1$

Combine $\frac{1}{2}$ and $2$ .

$\cos (θ) = \frac{3 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}} \cdot 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$\cos (θ) = \frac{3 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}} \cdot 1$

Rewrite the expression.

$\cos (θ) = \frac{3 \sqrt{3}}{2 \cdot 3^{1}} \cdot 1$

Evaluate the exponent.

$\cos (θ) = \frac{3 \sqrt{3}}{2 \cdot 3} \cdot 1$

Multiply $2$ by $3$ .

$\cos (θ) = \frac{3 \sqrt{3}}{6} \cdot 1$

Cancel the common factor of $3$ and $6$ .

Factor $3$ out of $3 \sqrt{3}$ .

$\cos (θ) = \frac{3 (\sqrt{3})}{6} \cdot 1$

Cancel the common factors.

Factor $3$ out of $6$ .

$\cos (θ) = \frac{3 \sqrt{3}}{3 \cdot 2} \cdot 1$

Cancel the common factor.

$\cos (θ) = \frac{3 \sqrt{3}}{3 \cdot 2} \cdot 1$

Rewrite the expression.

$\cos (θ) = \frac{\sqrt{3}}{2} \cdot 1$

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$\cos (θ) = \frac{\sqrt{3}}{2}$

Step 12

Take the inverse cosine of both sides of the equation to extract $θ$ from inside the cosine.

$θ = \arccos (\frac{\sqrt{3}}{2})$

Step 13

Simplify the right side.

The exact value of $\arccos (\frac{\sqrt{3}}{2})$ is $\frac{π}{6}$ .

$θ = \frac{π}{6}$

Step 14

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$θ = 2 π - \frac{π}{6}$

Step 15

Simplify $2 π - \frac{π}{6}$ .

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$θ = 2 π \cdot \frac{6}{6} - \frac{π}{6}$

Combine fractions.

Combine $2 π$ and $\frac{6}{6}$ .

$θ = \frac{2 π \cdot 6}{6} - \frac{π}{6}$

Combine the numerators over the common denominator.

$θ = \frac{2 π \cdot 6 - π}{6}$

Simplify the numerator.

Multiply $6$ by $2$ .

$θ = \frac{12 π - π}{6}$

Subtract $π$ from $12 π$ .

$θ = \frac{11 π}{6}$

Step 16

Find the period of $\cos (θ)$ .

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

Step 17

The period of the $\cos (θ)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$θ = \frac{π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Find All Complex Solutions tan(theta)=(2 square root of 3)/3sin(theta)

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