Trigonometry Examples

$\tan (150)$

Step 1

Find the value using the definition of tangent.

$\tan (150) = \frac{opposite}{adjacent}$

Step 2

Substitute the values into the definition.

$\tan (150) = \frac{\frac{1}{2}}{- \frac{\sqrt{3}}{2}}$

Step 3

Simplify the result.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} (- \frac{2}{\sqrt{3}})$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Move the leading negative in $- \frac{2}{\sqrt{3}}$ into the numerator.

$\frac{1}{2} \cdot \frac{- 2}{\sqrt{3}}$

Step 3.2.2

Factor $2$ out of $- 2$ .

$\frac{1}{2} \cdot \frac{2 (- 1)}{\sqrt{3}}$

Step 3.2.3

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2 \cdot - 1}{\sqrt{3}}$

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Move the negative in front of the fraction.

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

Step 3.4

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

$- (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

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

$- \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.5.2

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

$- \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.5.3

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

$- \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.5.4

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

$- \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.5.5

Add $1$ and $1$ .

$- \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.5.6

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

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Step 3.5.6.1

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

$- \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.5.6.2

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

$- \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.5.6.3

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

$- \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.6.4.1

Cancel the common factor.

$- \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.5.6.4.2

Rewrite the expression.

$- \frac{\sqrt{3}}{3^{1}}$

Step 3.5.6.5

Evaluate the exponent.

$- \frac{\sqrt{3}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{3}}{3}$

Decimal Form:

$- 0.57735026 \dots$

Step 5