Trigonometry Examples

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

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

Step 1

Simplify

\frac{1}{\sqrt{3}}

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

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Step 1.1

Multiply

\frac{1}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply

\frac{1}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

Step 1.2.2

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

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

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

Step 1.2.3

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

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

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

Step 1.2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 1.2.5

Add

1

$1$ and

1

$1$ .

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

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

Step 1.2.6

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 1.2.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

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

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

Step 1.2.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 1.2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 1.2.6.4

Cancel the common factor of

2

$2$ .

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Step 1.2.6.4.1

Cancel the common factor.

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

Step 1.2.6.4.2

Rewrite the expression.

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

Step 1.2.6.5

Evaluate the exponent.

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

Step 2

Take the inverse tangent of both sides of the equation to extract

$x$ from inside the tangent.

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

Step 3

Simplify the right side.

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Step 3.1

The exact value of

$\arctan (\frac{\sqrt{3}}{3})$ is

$\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 4

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.

$x = π + \frac{π}{6}$

Step 5

Simplify

$π + \frac{π}{6}$ .

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Step 5.1

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{6}{6}$ .

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

Step 5.2

Combine fractions.

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Step 5.2.1

Combine

$π$ and

$\frac{6}{6}$ .

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

Step 5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 + π}{6}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Move

$6$ to the left of

$π$ .

$x = \frac{6 \cdot π + π}{6}$

Step 5.3.2

Add

$6 π$ and

$π$ .

$x = \frac{7 π}{6}$

Step 6

Find the period of

$\tan (x)$ .

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Step 6.1

The period of the function can be calculated using

$\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 6.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 6.3

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

$0$ and

$1$ is

$1$ .

$\frac{π}{1}$

Step 6.4

Divide

$π$ by

$1$ .

$π$

Step 7

The period of the

$\tan (x)$ function is

$π$ so values will repeat every

$π$ radians in both directions.

$x = \frac{π}{6} + π n, \frac{7 π}{6} + π n$ , for any integer

$n$

Step 8

Consolidate the answers.

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

$n$