Trigonometry Examples

$250 - \frac{250}{\tan (60)} + \frac{500}{\sin (60)}$

Step 1

Simplify each term.

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Step 1.1

Separate fractions.

$250 - (\frac{250}{1} \cdot \frac{1}{\tan (60)}) + \frac{500}{\sin (60)}$

Step 1.2

Rewrite $\tan (60)$ in terms of sines and cosines.

$250 - (\frac{250}{1} \cdot \frac{1}{\frac{\sin (60)}{\cos (60)}}) + \frac{500}{\sin (60)}$

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (60)}{\cos (60)}$ .

$250 - (\frac{250}{1} (1 \frac{\cos (60)}{\sin (60)})) + \frac{500}{\sin (60)}$

Step 1.4

Write $1$ as a fraction with denominator $1$ .

$250 - (\frac{250}{1} (\frac{1}{1} \cdot \frac{\cos (60)}{\sin (60)})) + \frac{500}{\sin (60)}$

Step 1.5

Simplify.

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Step 1.5.1

Rewrite the expression.

$250 - (\frac{250}{1} (1 \frac{\cos (60)}{\sin (60)})) + \frac{500}{\sin (60)}$

Step 1.5.2

Multiply $\frac{\cos (60)}{\sin (60)}$ by $1$ .

$250 - (\frac{250}{1} \cdot \frac{\cos (60)}{\sin (60)}) + \frac{500}{\sin (60)}$

Step 1.6

Divide $250$ by $1$ .

$250 - (250 \frac{\cos (60)}{\sin (60)}) + \frac{500}{\sin (60)}$

Step 1.7

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$250 - (250 \frac{\frac{1}{2}}{\sin (60)}) + \frac{500}{\sin (60)}$

Step 1.8

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

$250 - (250 \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}) + \frac{500}{\sin (60)}$

Step 1.9

Multiply the numerator by the reciprocal of the denominator.

$250 - (250 (\frac{1}{2} \cdot \frac{2}{\sqrt{3}})) + \frac{500}{\sin (60)}$

Step 1.10

Cancel the common factor of $2$ .

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Step 1.10.1

Cancel the common factor.

$250 - (250 (\frac{1}{2} \cdot \frac{2}{\sqrt{3}})) + \frac{500}{\sin (60)}$

Step 1.10.2

Rewrite the expression.

$250 - (250 \frac{1}{\sqrt{3}}) + \frac{500}{\sin (60)}$

Step 1.11

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

$250 - (250 (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})) + \frac{500}{\sin (60)}$

Step 1.12

Combine and simplify the denominator.

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Step 1.12.1

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

$250 - (250 \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}) + \frac{500}{\sin (60)}$

Step 1.12.2

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

$250 - (250 \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}) + \frac{500}{\sin (60)}$

Step 1.12.3

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

$250 - (250 \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}) + \frac{500}{\sin (60)}$

Step 1.12.4

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

$250 - (250 \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}) + \frac{500}{\sin (60)}$

Step 1.12.5

Add $1$ and $1$ .

$250 - (250 \frac{\sqrt{3}}{{\sqrt{3}}^{2}}) + \frac{500}{\sin (60)}$

Step 1.12.6

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

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Step 1.12.6.1

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

$250 - (250 \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}) + \frac{500}{\sin (60)}$

Step 1.12.6.2

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

$250 - (250 \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}) + \frac{500}{\sin (60)}$

Step 1.12.6.3

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

$250 - (250 \frac{\sqrt{3}}{3^{\frac{2}{2}}}) + \frac{500}{\sin (60)}$

Step 1.12.6.4

Cancel the common factor of $2$ .

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Step 1.12.6.4.1

Cancel the common factor.

$250 - (250 \frac{\sqrt{3}}{3^{\frac{2}{2}}}) + \frac{500}{\sin (60)}$

Step 1.12.6.4.2

Rewrite the expression.

$250 - (250 \frac{\sqrt{3}}{3^{1}}) + \frac{500}{\sin (60)}$

Step 1.12.6.5

Evaluate the exponent.

$250 - (250 \frac{\sqrt{3}}{3}) + \frac{500}{\sin (60)}$

Step 1.13

Combine $250$ and $\frac{\sqrt{3}}{3}$ .

$250 - \frac{250 \sqrt{3}}{3} + \frac{500}{\sin (60)}$

Step 1.14

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

$250 - \frac{250 \sqrt{3}}{3} + \frac{500}{\frac{\sqrt{3}}{2}}$

Step 1.15

Multiply the numerator by the reciprocal of the denominator.

$250 - \frac{250 \sqrt{3}}{3} + 500 \frac{2}{\sqrt{3}}$

Step 1.16

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

$250 - \frac{250 \sqrt{3}}{3} + 500 (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 1.17

Combine and simplify the denominator.

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Step 1.17.1

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

$250 - \frac{250 \sqrt{3}}{3} + 500 \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 1.17.2

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

$250 - \frac{250 \sqrt{3}}{3} + 500 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 1.17.3

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

$250 - \frac{250 \sqrt{3}}{3} + 500 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 1.17.4

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

$250 - \frac{250 \sqrt{3}}{3} + 500 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 1.17.5

Add $1$ and $1$ .

$250 - \frac{250 \sqrt{3}}{3} + 500 \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 1.17.6

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

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Step 1.17.6.1

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

$250 - \frac{250 \sqrt{3}}{3} + 500 \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 1.17.6.2

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

$250 - \frac{250 \sqrt{3}}{3} + 500 \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 1.17.6.3

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

$250 - \frac{250 \sqrt{3}}{3} + 500 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.17.6.4

Cancel the common factor of $2$ .

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Step 1.17.6.4.1

Cancel the common factor.

$250 - \frac{250 \sqrt{3}}{3} + 500 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.17.6.4.2

Rewrite the expression.

$250 - \frac{250 \sqrt{3}}{3} + 500 \frac{2 \sqrt{3}}{3^{1}}$

Step 1.17.6.5

Evaluate the exponent.

$250 - \frac{250 \sqrt{3}}{3} + 500 \frac{2 \sqrt{3}}{3}$

Step 1.18

Multiply $500 \frac{2 \sqrt{3}}{3}$ .

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Step 1.18.1

Combine $500$ and $\frac{2 \sqrt{3}}{3}$ .

$250 - \frac{250 \sqrt{3}}{3} + \frac{500 (2 \sqrt{3})}{3}$

Step 1.18.2

Multiply $2$ by $500$ .

$250 - \frac{250 \sqrt{3}}{3} + \frac{1000 \sqrt{3}}{3}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$250 + \frac{- 250 \sqrt{3} + 1000 \sqrt{3}}{3}$

Step 2.2

Add $- 250 \sqrt{3}$ and $1000 \sqrt{3}$ .

$250 + \frac{750 \sqrt{3}}{3}$

Step 2.3

Cancel the common factor of $750$ and $3$ .

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Step 2.3.1

Factor $3$ out of $750 \sqrt{3}$ .

$250 + \frac{3 (250 \sqrt{3})}{3}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $3$ out of $3$ .

$250 + \frac{3 (250 \sqrt{3})}{3 (1)}$

Step 2.3.2.2

Cancel the common factor.

$250 + \frac{3 (250 \sqrt{3})}{3 \cdot 1}$

Step 2.3.2.3

Rewrite the expression.

$250 + \frac{250 \sqrt{3}}{1}$

Step 2.3.2.4

Divide $250 \sqrt{3}$ by $1$ .

$250 + 250 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$250 + 250 \sqrt{3}$

Decimal Form:

$683.01270189 \dots$