Trigonometry Examples

$\frac{4 \sin (45) \cos (45)}{\tan (30) \cot (30)}$

Step 1

Simplify the numerator.

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Step 1.1

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{4 \frac{\sqrt{2}}{2} \cos (45)}{\tan (30) \cot (30)}$

Step 1.2

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{4 \frac{\sqrt{2}}{2} \frac{\sqrt{2}}{2}}{\tan (30) \cot (30)}$

Step 1.3

Combine exponents.

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Step 1.3.1

Combine $4$ and $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{4 \sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}}{\tan (30) \cot (30)}$

Step 1.3.2

Multiply $\frac{4 \sqrt{2}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{4 \sqrt{2} \sqrt{2}}{2 \cdot 2}}{\tan (30) \cot (30)}$

Step 1.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\frac{4 ({\sqrt{2}}^{1} \sqrt{2})}{2 \cdot 2}}{\tan (30) \cot (30)}$

Step 1.3.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\frac{4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2 \cdot 2}}{\tan (30) \cot (30)}$

Step 1.3.5

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

$\frac{\frac{4 {\sqrt{2}}^{1 + 1}}{2 \cdot 2}}{\tan (30) \cot (30)}$

Step 1.3.6

Add $1$ and $1$ .

$\frac{\frac{4 {\sqrt{2}}^{2}}{2 \cdot 2}}{\tan (30) \cot (30)}$

Step 1.3.7

Multiply $2$ by $2$ .

$\frac{\frac{4 {\sqrt{2}}^{2}}{4}}{\tan (30) \cot (30)}$

Step 1.4

Reduce the expression by cancelling the common factors.

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Step 1.4.1

Reduce the expression $\frac{4 {\sqrt{2}}^{2}}{4}$ by cancelling the common factors.

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Step 1.4.1.1

Cancel the common factor.

$\frac{\frac{4 {\sqrt{2}}^{2}}{4}}{\tan (30) \cot (30)}$

Step 1.4.1.2

Rewrite the expression.

$\frac{\frac{{\sqrt{2}}^{2}}{1}}{\tan (30) \cot (30)}$

Step 1.4.2

Divide ${\sqrt{2}}^{2}$ by $1$ .

$\frac{{\sqrt{2}}^{2}}{\tan (30) \cot (30)}$

Step 1.5

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

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Step 1.5.1

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

$\frac{{(2^{\frac{1}{2}})}^{2}}{\tan (30) \cot (30)}$

Step 1.5.2

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

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

Step 1.5.3

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

$\frac{2^{\frac{2}{2}}}{\tan (30) \cot (30)}$

Step 1.5.4

Cancel the common factor of $2$ .

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Step 1.5.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{\tan (30) \cot (30)}$

Step 1.5.4.2

Rewrite the expression.

$\frac{2^{1}}{\tan (30) \cot (30)}$

Step 1.5.5

Evaluate the exponent.

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

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{2}{\frac{\sqrt{3}}{3} \cot (30)}$

Step 2.2

The exact value of $\cot (30)$ is $\sqrt{3}$ .

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

Step 3

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

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 5

Simplify by cancelling exponent with radical.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Cancel the common factor of $2$ .

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Step 5.1.4.1

Cancel the common factor.

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

Step 5.1.4.2

Rewrite the expression.

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

Step 5.1.5

Evaluate the exponent.

$\frac{2}{\frac{3}{3}}$

Step 5.2

Simplify the expression.

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Step 5.2.1

Divide $3$ by $3$ .

$\frac{2}{1}$

Step 5.2.2

Divide $2$ by $1$ .

$2$