Basic Math Examples

$\frac{{(\tan (60) - \cos (30))}^{2}}{\sin (60)}$

Step 1

Simplify the numerator.

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Step 1.1

The exact value of $\tan (60)$ is $\sqrt{3}$ .

$\frac{{(\sqrt{3} - \cos (30))}^{2}}{\sin (60)}$

Step 1.2

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

$\frac{{(\sqrt{3} - \frac{\sqrt{3}}{2})}^{2}}{\sin (60)}$

Step 1.3

To write $\sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{{(\sqrt{3} \cdot \frac{2}{2} - \frac{\sqrt{3}}{2})}^{2}}{\sin (60)}$

Step 1.4

Combine $\sqrt{3}$ and $\frac{2}{2}$ .

$\frac{{(\frac{\sqrt{3} \cdot 2}{2} - \frac{\sqrt{3}}{2})}^{2}}{\sin (60)}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{{(\frac{\sqrt{3} \cdot 2 - \sqrt{3}}{2})}^{2}}{\sin (60)}$

Step 1.6

Rewrite $\frac{\sqrt{3} \cdot 2 - \sqrt{3}}{2}$ in a factored form.

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Step 1.6.1

Move $2$ to the left of $\sqrt{3}$ .

$\frac{{(\frac{2 \cdot \sqrt{3} - \sqrt{3}}{2})}^{2}}{\sin (60)}$

Step 1.6.2

Subtract $\sqrt{3}$ from $2 \sqrt{3}$ .

$\frac{{(\frac{\sqrt{3}}{2})}^{2}}{\sin (60)}$

Step 1.7

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

$\frac{\frac{{\sqrt{3}}^{2}}{2^{2}}}{\sin (60)}$

Step 1.8

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

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Step 1.8.1

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

$\frac{\frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}}{\sin (60)}$

Step 1.8.2

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

$\frac{\frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}}{\sin (60)}$

Step 1.8.3

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

$\frac{\frac{3^{\frac{2}{2}}}{2^{2}}}{\sin (60)}$

Step 1.8.4

Cancel the common factor of $2$ .

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Step 1.8.4.1

Cancel the common factor.

$\frac{\frac{3^{\frac{2}{2}}}{2^{2}}}{\sin (60)}$

Step 1.8.4.2

Rewrite the expression.

$\frac{\frac{3^{1}}{2^{2}}}{\sin (60)}$

Step 1.8.5

Evaluate the exponent.

$\frac{\frac{3}{2^{2}}}{\sin (60)}$

Step 1.9

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

$\frac{\frac{3}{4}}{\sin (60)}$

Step 2

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{3}{4} \cdot \frac{2}{\sqrt{3}}$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Factor $2$ out of $4$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Multiply $\frac{3}{2}$ by $\frac{1}{\sqrt{3}}$ .

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

Step 6

Multiply $\frac{3}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{3}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 7.2

Move $\sqrt{3}$ .

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 7.6

Add $1$ and $1$ .

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

Step 7.7

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

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Step 7.7.1

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

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

Step 7.7.2

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

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

Step 7.7.3

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

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

Step 7.7.4

Cancel the common factor of $2$ .

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Step 7.7.4.1

Cancel the common factor.

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

Step 7.7.4.2

Rewrite the expression.

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

Step 7.7.5

Evaluate the exponent.

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

Step 8

Cancel the common factor of $3$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.86602540 \dots$