Trigonometry Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify the left side.

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Step 1.2.1

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

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Rewrite the expression.

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

Step 1.2.2

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

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Step 1.2.2.1

Cancel the common factor.

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

Step 1.2.2.2

Rewrite the expression.

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Separate fractions.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 1.3.5

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

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Step 1.3.5.1

Cancel the common factor.

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

Step 1.3.5.2

Rewrite the expression.

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

Step 1.3.6

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

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

Step 1.3.7

Combine and simplify the denominator.

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Step 1.3.7.1

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

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

Step 1.3.7.2

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

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

Step 1.3.7.3

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

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

Step 1.3.7.4

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

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

Step 1.3.7.5

Add $1$ and $1$ .

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

Step 1.3.7.6

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

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Step 1.3.7.6.1

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

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

Step 1.3.7.6.2

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

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

Step 1.3.7.6.3

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

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

Step 1.3.7.6.4

Cancel the common factor of $2$ .

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Step 1.3.7.6.4.1

Cancel the common factor.

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

Step 1.3.7.6.4.2

Rewrite the expression.

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

Step 1.3.7.6.5

Evaluate the exponent.

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

Step 1.3.8

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

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

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Step 4.1.1.1

Cancel the common factor of $3$ .

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Step 4.1.1.1.1

Cancel the common factor.

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

Step 4.1.1.1.2

Rewrite the expression.

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

Step 4.1.1.2

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

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Step 4.1.1.2.1

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

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

Step 4.1.1.2.2

Cancel the common factor.

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

Step 4.1.1.2.3

Rewrite the expression.

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

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

Combine and simplify the denominator.

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Step 4.2.1.2.1

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

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

Step 4.2.1.2.2

Move $\sqrt{3}$ .

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

Step 4.2.1.2.3

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

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

Step 4.2.1.2.4

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

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

Step 4.2.1.2.5

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

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

Step 4.2.1.2.6

Add $1$ and $1$ .

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

Step 4.2.1.2.7

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

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Step 4.2.1.2.7.1

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

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

Step 4.2.1.2.7.2

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

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

Step 4.2.1.2.7.3

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

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

Step 4.2.1.2.7.4

Cancel the common factor of $2$ .

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Step 4.2.1.2.7.4.1

Cancel the common factor.

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

Step 4.2.1.2.7.4.2

Rewrite the expression.

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

Step 4.2.1.2.7.5

Evaluate the exponent.

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

Step 4.2.1.3

Multiply $2$ by $3$ .

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

Step 4.2.1.4

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

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Step 4.2.1.4.1

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

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

Step 4.2.1.4.2

Cancel the common factors.

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Step 4.2.1.4.2.1

Factor $3$ out of $6$ .

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

Step 4.2.1.4.2.2

Cancel the common factor.

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

Step 4.2.1.4.2.3

Rewrite the expression.

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

Step 4.2.1.5

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

$\cos (x) = \frac{\sqrt{3}}{2}$

Step 5

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

$x = \arccos (\frac{\sqrt{3}}{2})$

Step 6

Simplify the right side.

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Step 6.1

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

$x = \frac{π}{6}$

Step 7

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.

$x = 2 π - \frac{π}{6}$

Step 8

Simplify $2 π - \frac{π}{6}$ .

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Step 8.1

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

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

Step 8.2

Combine fractions.

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Step 8.2.1

Combine $2 π$ and $\frac{6}{6}$ .

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

Step 8.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - π}{6}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - π}{6}$

Step 8.3.2

Subtract $π$ from $12 π$ .

$x = \frac{11 π}{6}$

Step 9

Find the period of $\cos (x)$ .

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Step 9.1

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

$\frac{2 π}{| b |}$

Step 9.2

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

$\frac{2 π}{| 1 |}$

Step 9.3

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

$\frac{2 π}{1}$

Step 9.4

Divide $2 π$ by $1$ .

$2 π$

Step 10

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

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