Basic Math Examples

$\frac{\sin (30) \cdot \cos (60)}{\tan (60)} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1

Simplify each term.

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Step 1.1

Separate fractions.

$\frac{\cos (60)}{1} \cdot \frac{\sin (30)}{\tan (60)} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.2

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

$\frac{\cos (60)}{1} \cdot \frac{\sin (30)}{\frac{\sin (60)}{\cos (60)}} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.3

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

$\frac{\cos (60)}{1} (\sin (30) \frac{\cos (60)}{\sin (60)}) + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.4

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

$\frac{\cos (60)}{1} (\frac{\sin (30)}{1} \cdot \frac{\cos (60)}{\sin (60)}) + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.5

Simplify.

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Step 1.5.1

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

$\frac{\cos (60)}{1} (\sin (30) \frac{\cos (60)}{\sin (60)}) + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.5.2

Combine $\sin (30)$ and $\frac{\cos (60)}{\sin (60)}$ .

$\frac{\cos (60)}{1} \cdot \frac{\sin (30) \cos (60)}{\sin (60)} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.6

Divide $\cos (60)$ by $1$ .

$\cos (60) \frac{\sin (30) \cos (60)}{\sin (60)} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.7

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

$\frac{1}{2} \cdot \frac{\sin (30) \cos (60)}{\sin (60)} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.8

Simplify the numerator.

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Step 1.8.1

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

$\frac{1}{2} \cdot \frac{\frac{1}{2} \cos (60)}{\sin (60)} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.8.2

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

$\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \frac{1}{2}}{\sin (60)} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.9

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

$\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \frac{1}{2}}{\frac{\sqrt{3}}{2}} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.10

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

$\frac{1}{2} \cdot \frac{\frac{1}{2 \cdot 2}}{\frac{\sqrt{3}}{2}} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.11

Multiply $2$ by $2$ .

$\frac{1}{2} \cdot \frac{\frac{1}{4}}{\frac{\sqrt{3}}{2}} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.12

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} (\frac{1}{4} \cdot \frac{2}{\sqrt{3}}) + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.13

Cancel the common factor of $2$ .

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Step 1.13.1

Factor $2$ out of $4$ .

$\frac{1}{2} (\frac{1}{2 (2)} \cdot \frac{2}{\sqrt{3}}) + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.13.2

Cancel the common factor.

$\frac{1}{2} (\frac{1}{2 \cdot 2} \cdot \frac{2}{\sqrt{3}}) + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.13.3

Rewrite the expression.

$\frac{1}{2} (\frac{1}{2} \cdot \frac{1}{\sqrt{3}}) + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.14

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

$\frac{1}{2} \cdot \frac{1}{2 \sqrt{3}} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.15

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

$\frac{1}{2} (\frac{1}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.16

Combine and simplify the denominator.

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Step 1.16.1

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

$\frac{1}{2} \cdot \frac{\sqrt{3}}{2 \sqrt{3} \sqrt{3}} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.16.2

Move $\sqrt{3}$ .

$\frac{1}{2} \cdot \frac{\sqrt{3}}{2 (\sqrt{3} \sqrt{3})} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.16.3

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

$\frac{1}{2} \cdot \frac{\sqrt{3}}{2 ({\sqrt{3}}^{1} \sqrt{3})} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.16.4

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

$\frac{1}{2} \cdot \frac{\sqrt{3}}{2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.16.5

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

$\frac{1}{2} \cdot \frac{\sqrt{3}}{2 {\sqrt{3}}^{1 + 1}} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.16.6

Add $1$ and $1$ .

$\frac{1}{2} \cdot \frac{\sqrt{3}}{2 {\sqrt{3}}^{2}} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.16.7

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

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Step 1.16.7.1

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

$\frac{1}{2} \cdot \frac{\sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.16.7.2

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

$\frac{1}{2} \cdot \frac{\sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.16.7.3

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

$\frac{1}{2} \cdot \frac{\sqrt{3}}{2 \cdot 3^{\frac{2}{2}}} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.16.7.4

Cancel the common factor of $2$ .

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Step 1.16.7.4.1

Cancel the common factor.

$\frac{1}{2} \cdot \frac{\sqrt{3}}{2 \cdot 3^{\frac{2}{2}}} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.16.7.4.2

Rewrite the expression.

$\frac{1}{2} \cdot \frac{\sqrt{3}}{2 \cdot 3^{1}} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.16.7.5

Evaluate the exponent.

$\frac{1}{2} \cdot \frac{\sqrt{3}}{2 \cdot 3} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.17

Multiply $2$ by $3$ .

$\frac{1}{2} \cdot \frac{\sqrt{3}}{6} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.18

Multiply $\frac{1}{2} \cdot \frac{\sqrt{3}}{6}$ .

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Step 1.18.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{3}}{6}$ .

$\frac{\sqrt{3}}{2 \cdot 6} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.18.2

Multiply $2$ by $6$ .

$\frac{\sqrt{3}}{12} + \frac{\cos (30) \cdot \sin (60)}{\cot (30)}$

Step 1.19

Separate fractions.

$\frac{\sqrt{3}}{12} + \frac{\sin (60)}{1} \cdot \frac{\cos (30)}{\cot (30)}$

Step 1.20

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

$\frac{\sqrt{3}}{12} + \frac{\sin (60)}{1} \cdot \frac{\cos (30)}{\frac{\cos (30)}{\sin (30)}}$

Step 1.21

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

$\frac{\sqrt{3}}{12} + \frac{\sin (60)}{1} (\cos (30) \frac{\sin (30)}{\cos (30)})$

Step 1.22

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

$\frac{\sqrt{3}}{12} + \frac{\sin (60)}{1} (\frac{\cos (30)}{1} \cdot \frac{\sin (30)}{\cos (30)})$

Step 1.23

Cancel the common factor of $\cos (30)$ .

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Step 1.23.1

Cancel the common factor.

$\frac{\sqrt{3}}{12} + \frac{\sin (60)}{1} (\frac{\cos (30)}{1} \cdot \frac{\sin (30)}{\cos (30)})$

Step 1.23.2

Rewrite the expression.

$\frac{\sqrt{3}}{12} + \frac{\sin (60)}{1} \sin (30)$

Step 1.24

Divide $\sin (60)$ by $1$ .

$\frac{\sqrt{3}}{12} + \sin (60) \sin (30)$

Step 1.25

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

$\frac{\sqrt{3}}{12} + \frac{\sqrt{3}}{2} \sin (30)$

Step 1.26

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

$\frac{\sqrt{3}}{12} + \frac{\sqrt{3}}{2} \cdot \frac{1}{2}$

Step 1.27

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

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Step 1.27.1

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

$\frac{\sqrt{3}}{12} + \frac{\sqrt{3}}{2 \cdot 2}$

Step 1.27.2

Multiply $2$ by $2$ .

$\frac{\sqrt{3}}{12} + \frac{\sqrt{3}}{4}$

Step 2

To write $\frac{\sqrt{3}}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\sqrt{3}}{12} + \frac{\sqrt{3}}{4} \cdot \frac{3}{3}$

Step 3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{\sqrt{3}}{4}$ by $\frac{3}{3}$ .

$\frac{\sqrt{3}}{12} + \frac{\sqrt{3} \cdot 3}{4 \cdot 3}$

Step 3.2

Multiply $4$ by $3$ .

$\frac{\sqrt{3}}{12} + \frac{\sqrt{3} \cdot 3}{12}$

Step 4

Combine the numerators over the common denominator.

$\frac{\sqrt{3} + \sqrt{3} \cdot 3}{12}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{\sqrt{3} + 3 \cdot \sqrt{3}}{12}$

Step 5.2

Add $\sqrt{3}$ and $3 \sqrt{3}$ .

$\frac{4 \sqrt{3}}{12}$

Step 6

Cancel the common factor of $4$ and $12$ .

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Step 6.1

Factor $4$ out of $4 \sqrt{3}$ .

$\frac{4 (\sqrt{3})}{12}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $4$ out of $12$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.57735026 \dots$