Trigonometry Examples

$\frac{2 \cos (30)}{\tan (30)}$

Step 1

Separate fractions.

$\frac{2}{1} \cdot \frac{\cos (30)}{\tan (30)}$

Step 2

Rewrite $\tan (30)$ in terms of sines and cosines.

$\frac{2}{1} \cdot \frac{\cos (30)}{\frac{\sin (30)}{\cos (30)}}$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (30)}{\cos (30)}$ .

$\frac{2}{1} (\cos (30) \frac{\cos (30)}{\sin (30)})$

Step 4

Write $\cos (30)$ as a fraction with denominator $1$ .

$\frac{2}{1} (\frac{\cos (30)}{1} \cdot \frac{\cos (30)}{\sin (30)})$

Step 5

Simplify.

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Step 5.1

Divide $\cos (30)$ by $1$ .

$\frac{2}{1} (\cos (30) \frac{\cos (30)}{\sin (30)})$

Step 5.2

Combine $\cos (30)$ and $\frac{\cos (30)}{\sin (30)}$ .

$\frac{2}{1} \cdot \frac{\cos (30) \cos (30)}{\sin (30)}$

Step 6

Divide $2$ by $1$ .

$2 \frac{\cos (30) \cos (30)}{\sin (30)}$

Step 7

Simplify the numerator.

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Step 7.1

Raise $\cos (30)$ to the power of $1$ .

$2 \frac{\cos^{1} (30) \cos (30)}{\sin (30)}$

Step 7.2

Raise $\cos (30)$ to the power of $1$ .

$2 \frac{\cos^{1} (30) \cos^{1} (30)}{\sin (30)}$

Step 7.3

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

$2 \frac{{\cos (30)}^{1 + 1}}{\sin (30)}$

Step 7.4

Add $1$ and $1$ .

$2 \frac{\cos^{2} (30)}{\sin (30)}$

Step 8

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$2 \frac{\cos^{2} (30)}{\frac{1}{2}}$

Step 9

Simplify the numerator.

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Step 9.1

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

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

Step 9.2

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

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

Step 9.3

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

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Step 9.3.1

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

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

Step 9.3.2

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

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

Step 9.3.3

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

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

Step 9.3.4

Cancel the common factor of $2$ .

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Step 9.3.4.1

Cancel the common factor.

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

Step 9.3.4.2

Rewrite the expression.

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

Step 9.3.5

Evaluate the exponent.

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

Step 9.4

Raise $2$ to the power of $2$ .

$2 \frac{\frac{3}{4}}{\frac{1}{2}}$

Step 10

Multiply the numerator by the reciprocal of the denominator.

$2 (\frac{3}{4} \cdot 2)$

Step 11

Cancel the common factor of $2$ .

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Step 11.1

Factor $2$ out of $4$ .

$2 (\frac{3}{2 (2)} \cdot 2)$

Step 11.2

Cancel the common factor.

$2 (\frac{3}{2 \cdot 2} \cdot 2)$

Step 11.3

Rewrite the expression.

$2 (\frac{3}{2})$

Step 12

Cancel the common factor of $2$ .

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Step 12.1

Cancel the common factor.

$2 (\frac{3}{2})$

Step 12.2

Rewrite the expression.

$3$