Trigonometry Examples

$2 \sqrt{3} \cos (x) \csc (x) + 4 \cos (x) - 3 \csc (x) - 2 \sqrt{3}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\csc (x)$ in terms of sines and cosines.

$2 \sqrt{3} \cos (x) \frac{1}{\sin (x)} + 4 \cos (x) - 3 \csc (x) - 2 \sqrt{3}$

Step 1.2

Multiply $2 \sqrt{3} \cos (x) \frac{1}{\sin (x)}$ .

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Step 1.2.1

Combine $\frac{1}{\sin (x)}$ and $2$ .

$\frac{2}{\sin (x)} \sqrt{3} \cos (x) + 4 \cos (x) - 3 \csc (x) - 2 \sqrt{3}$

Step 1.2.2

Combine $\frac{2}{\sin (x)}$ and $\sqrt{3}$ .

$\frac{2 \sqrt{3}}{\sin (x)} \cos (x) + 4 \cos (x) - 3 \csc (x) - 2 \sqrt{3}$

Step 1.2.3

Combine $\frac{2 \sqrt{3}}{\sin (x)}$ and $\cos (x)$ .

$\frac{2 \sqrt{3} \cos (x)}{\sin (x)} + 4 \cos (x) - 3 \csc (x) - 2 \sqrt{3}$

Step 1.3

Rewrite $\csc (x)$ in terms of sines and cosines.

$\frac{2 \sqrt{3} \cos (x)}{\sin (x)} + 4 \cos (x) - 3 \frac{1}{\sin (x)} - 2 \sqrt{3}$

Step 1.4

Combine $- 3$ and $\frac{1}{\sin (x)}$ .

$\frac{2 \sqrt{3} \cos (x)}{\sin (x)} + 4 \cos (x) + \frac{- 3}{\sin (x)} - 2 \sqrt{3}$

Step 1.5

Move the negative in front of the fraction.

$\frac{2 \sqrt{3} \cos (x)}{\sin (x)} + 4 \cos (x) - \frac{3}{\sin (x)} - 2 \sqrt{3}$

Step 2

Simplify each term.

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Step 2.1

Separate fractions.

$\frac{2 \sqrt{3}}{1} \cdot \frac{\cos (x)}{\sin (x)} + 4 \cos (x) - \frac{3}{\sin (x)} - 2 \sqrt{3}$

Step 2.2

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\frac{2 \sqrt{3}}{1} \cot (x) + 4 \cos (x) - \frac{3}{\sin (x)} - 2 \sqrt{3}$

Step 2.3

Divide $2 \sqrt{3}$ by $1$ .

$2 \sqrt{3} \cot (x) + 4 \cos (x) - \frac{3}{\sin (x)} - 2 \sqrt{3}$

Step 2.4

Separate fractions.

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

Step 2.5

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$2 \sqrt{3} \cot (x) + 4 \cos (x) - (\frac{3}{1} \csc (x)) - 2 \sqrt{3}$

Step 2.6

Divide $3$ by $1$ .

$2 \sqrt{3} \cot (x) + 4 \cos (x) - (3 \csc (x)) - 2 \sqrt{3}$

Step 2.7

Multiply $3$ by $- 1$ .

$2 \sqrt{3} \cot (x) + 4 \cos (x) - 3 \csc (x) - 2 \sqrt{3}$

Step 3

Simplify each term.

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Step 3.1

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

$2 \sqrt{3} \frac{\cos (x)}{\sin (x)} + 4 \cos (x) - 3 \csc (x) - 2 \sqrt{3}$

Step 3.2

Multiply $2 \sqrt{3} \frac{\cos (x)}{\sin (x)}$ .

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Step 3.2.1

Combine $\frac{\cos (x)}{\sin (x)}$ and $2$ .

$\frac{\cos (x) \cdot 2}{\sin (x)} \sqrt{3} + 4 \cos (x) - 3 \csc (x) - 2 \sqrt{3}$

Step 3.2.2

Combine $\frac{\cos (x) \cdot 2}{\sin (x)}$ and $\sqrt{3}$ .

$\frac{\cos (x) \cdot 2 \sqrt{3}}{\sin (x)} + 4 \cos (x) - 3 \csc (x) - 2 \sqrt{3}$

Step 3.3

Move $2$ to the left of $\cos (x)$ .

$\frac{2 \cdot \cos (x) \sqrt{3}}{\sin (x)} + 4 \cos (x) - 3 \csc (x) - 2 \sqrt{3}$

Step 3.4

Rewrite $\csc (x)$ in terms of sines and cosines.

$\frac{2 \cos (x) \sqrt{3}}{\sin (x)} + 4 \cos (x) - 3 \frac{1}{\sin (x)} - 2 \sqrt{3}$

Step 3.5

Combine $- 3$ and $\frac{1}{\sin (x)}$ .

$\frac{2 \cos (x) \sqrt{3}}{\sin (x)} + 4 \cos (x) + \frac{- 3}{\sin (x)} - 2 \sqrt{3}$

Step 3.6

Move the negative in front of the fraction.

$\frac{2 \cos (x) \sqrt{3}}{\sin (x)} + 4 \cos (x) - \frac{3}{\sin (x)} - 2 \sqrt{3}$

Step 4

Simplify each term.

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Step 4.1

Separate fractions.

$\frac{2 \sqrt{3}}{1} \cdot \frac{\cos (x)}{\sin (x)} + 4 \cos (x) - \frac{3}{\sin (x)} - 2 \sqrt{3}$

Step 4.2

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\frac{2 \sqrt{3}}{1} \cot (x) + 4 \cos (x) - \frac{3}{\sin (x)} - 2 \sqrt{3}$

Step 4.3

Divide $2 \sqrt{3}$ by $1$ .

$2 \sqrt{3} \cot (x) + 4 \cos (x) - \frac{3}{\sin (x)} - 2 \sqrt{3}$

Step 4.4

Separate fractions.

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

Step 4.5

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$2 \sqrt{3} \cot (x) + 4 \cos (x) - (\frac{3}{1} \csc (x)) - 2 \sqrt{3}$

Step 4.6

Divide $3$ by $1$ .

$2 \sqrt{3} \cot (x) + 4 \cos (x) - (3 \csc (x)) - 2 \sqrt{3}$

Step 4.7

Multiply $3$ by $- 1$ .

$2 \sqrt{3} \cot (x) + 4 \cos (x) - 3 \csc (x) - 2 \sqrt{3}$