Trigonometry Examples

$\sqrt{3} \tan (a) - 2 \sin (b) \tan (a) = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\sqrt{3} \tan (a) - 2 \sin (b) \tan (a)$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

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

$\sqrt{3} \frac{\sin (a)}{\cos (a)} - 2 \sin (b) \tan (a) = 0$

Step 1.1.1.2

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

$\frac{\sqrt{3} \sin (a)}{\cos (a)} - 2 \sin (b) \tan (a) = 0$

Step 1.1.1.3

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

$\frac{\sqrt{3} \sin (a)}{\cos (a)} - 2 \sin (b) \frac{\sin (a)}{\cos (a)} = 0$

Step 1.1.1.4

Multiply $- 2 \sin (b) \frac{\sin (a)}{\cos (a)}$ .

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Step 1.1.1.4.1

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

$\frac{\sqrt{3} \sin (a)}{\cos (a)} + \frac{\sin (a) \cdot - 2}{\cos (a)} \sin (b) = 0$

Step 1.1.1.4.2

Combine $\frac{\sin (a) \cdot - 2}{\cos (a)}$ and $\sin (b)$ .

$\frac{\sqrt{3} \sin (a)}{\cos (a)} + \frac{\sin (a) \cdot - 2 \sin (b)}{\cos (a)} = 0$

Step 1.1.1.5

Move $- 2$ to the left of $\sin (a)$ .

$\frac{\sqrt{3} \sin (a)}{\cos (a)} + \frac{- 2 \cdot \sin (a) \sin (b)}{\cos (a)} = 0$

Step 1.1.1.6

Move the negative in front of the fraction.

$\frac{\sqrt{3} \sin (a)}{\cos (a)} - \frac{2 \sin (a) \sin (b)}{\cos (a)} = 0$

Step 1.1.2

Simplify each term.

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Step 1.1.2.1

Separate fractions.

$\frac{\sqrt{3}}{1} \cdot \frac{\sin (a)}{\cos (a)} - \frac{2 \sin (a) \sin (b)}{\cos (a)} = 0$

Step 1.1.2.2

Convert from $\frac{\sin (a)}{\cos (a)}$ to $\tan (a)$ .

$\frac{\sqrt{3}}{1} \tan (a) - \frac{2 \sin (a) \sin (b)}{\cos (a)} = 0$

Step 1.1.2.3

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

$\sqrt{3} \tan (a) - \frac{2 \sin (a) \sin (b)}{\cos (a)} = 0$

Step 1.1.2.4

Separate fractions.

$\sqrt{3} \tan (a) - (\frac{2 \sin (b)}{1} \cdot \frac{\sin (a)}{\cos (a)}) = 0$

Step 1.1.2.5

Convert from $\frac{\sin (a)}{\cos (a)}$ to $\tan (a)$ .

$\sqrt{3} \tan (a) - (\frac{2 \sin (b)}{1} \tan (a)) = 0$

Step 1.1.2.6

Divide $2 \sin (b)$ by $1$ .

$\sqrt{3} \tan (a) - (2 \sin (b) \tan (a)) = 0$

Step 1.1.2.7

Multiply $2$ by $- 1$ .

$\sqrt{3} \tan (a) - 2 \sin (b) \tan (a) = 0$

Step 2

Subtract $\sqrt{3} \tan (a)$ from both sides of the equation.

$- 2 \sin (b) \tan (a) = - \sqrt{3} \tan (a)$