Precalculus Examples

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

Step 1

Divide each term in the equation by $\cos (2 x)$ .

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

Step 2

Separate fractions.

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

Step 3

Convert from $\frac{\sin (2 x)}{\cos (2 x)}$ to $\tan (2 x)$ .

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

Step 4

Divide $\sqrt{3}$ by $1$ .

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

Step 5

Cancel the common factor of $\cos (2 x)$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Divide each term in $\sqrt{3} \tan (2 x) = 1$ by $\sqrt{3}$ and simplify.

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Step 6.1

Divide each term in $\sqrt{3} \tan (2 x) = 1$ by $\sqrt{3}$ .

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

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $\tan (2 x)$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

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

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

Step 6.3.2

Combine and simplify the denominator.

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Step 6.3.2.1

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

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

Step 6.3.2.2

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

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

Step 6.3.2.3

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

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

Step 6.3.2.4

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

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

Step 6.3.2.5

Add $1$ and $1$ .

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

Step 6.3.2.6

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

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Step 6.3.2.6.1

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

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

Step 6.3.2.6.2

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

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

Step 6.3.2.6.3

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

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

Step 6.3.2.6.4

Cancel the common factor of $2$ .

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Step 6.3.2.6.4.1

Cancel the common factor.

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

Step 6.3.2.6.4.2

Rewrite the expression.

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

Step 6.3.2.6.5

Evaluate the exponent.

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

Step 7

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

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

Step 8

Simplify the right side.

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Step 8.1

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

$2 x = \frac{π}{6}$

Step 9

Divide each term in $2 x = \frac{π}{6}$ by $2$ and simplify.

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Step 9.1

Divide each term in $2 x = \frac{π}{6}$ by $2$ .

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

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $2$ .

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Step 9.2.1.1

Cancel the common factor.

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

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{6}}{2}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{6} \cdot \frac{1}{2}$

Step 9.3.2

Multiply $\frac{π}{6} \cdot \frac{1}{2}$ .

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Step 9.3.2.1

Multiply $\frac{π}{6}$ by $\frac{1}{2}$ .

$x = \frac{π}{6 \cdot 2}$

Step 9.3.2.2

Multiply $6$ by $2$ .

$x = \frac{π}{12}$

Step 10

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$2 x = π + \frac{π}{6}$

Step 11

Solve for $x$ .

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Step 11.1

Simplify.

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Step 11.1.1

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

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

Step 11.1.2

Combine $π$ and $\frac{6}{6}$ .

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

Step 11.1.3

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 6 + π}{6}$

Step 11.1.4

Add $π \cdot 6$ and $π$ .

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Step 11.1.4.1

Reorder $π$ and $6$ .

$2 x = \frac{6 \cdot π + π}{6}$

Step 11.1.4.2

Add $6 \cdot π$ and $π$ .

$2 x = \frac{7 \cdot π}{6}$

Step 11.2

Divide each term in $2 x = \frac{7 \cdot π}{6}$ by $2$ and simplify.

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Step 11.2.1

Divide each term in $2 x = \frac{7 \cdot π}{6}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{7 \cdot π}{6}}{2}$

Step 11.2.2

Simplify the left side.

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Step 11.2.2.1

Cancel the common factor of $2$ .

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Step 11.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{7 \cdot π}{6}}{2}$

Step 11.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{7 \cdot π}{6}}{2}$

Step 11.2.3

Simplify the right side.

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Step 11.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 \cdot π}{6} \cdot \frac{1}{2}$

Step 11.2.3.2

Multiply $\frac{7 π}{6} \cdot \frac{1}{2}$ .

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Step 11.2.3.2.1

Multiply $\frac{7 π}{6}$ by $\frac{1}{2}$ .

$x = \frac{7 π}{6 \cdot 2}$

Step 11.2.3.2.2

Multiply $6$ by $2$ .

$x = \frac{7 π}{12}$

Step 12

Find the period of $\tan (2 x)$ .

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Step 12.1

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

$\frac{π}{| b |}$

Step 12.2

Replace $b$ with $2$ in the formula for period.

$\frac{π}{| 2 |}$

Step 12.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{π}{2}$

Step 13

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

$x = \frac{π}{12} + \frac{π n}{2}, \frac{7 π}{12} + \frac{π n}{2}$ , for any integer $n$

Step 14

Consolidate the answers.

$x = \frac{π}{12} + \frac{π n}{2}$ , for any integer $n$