Algebra Examples

$\tan (x + π) + 2 \sin (x + π) = 0$

Step 1

Simplify $\tan (x + π) + 2 \sin (x + π)$ .

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Step 1.1

Rewrite $\tan (x + π)$ in terms of sines and cosines.

$\frac{\sin (x + π)}{\cos (x + π)} + 2 \sin (x + π) = 0$

Step 1.2

To write $2 \sin (x + π)$ as a fraction with a common denominator, multiply by $\frac{\cos (x + π)}{\cos (x + π)}$ .

$\frac{\sin (x + π)}{\cos (x + π)} + \frac{2 \sin (x + π) \cos (x + π)}{\cos (x + π)} = 0$

Step 1.3

Combine the numerators over the common denominator.

$\frac{\sin (x + π) + 2 \sin (x + π) \cos (x + π)}{\cos (x + π)} = 0$

Step 1.4

Factor $\sin (x + π)$ out of $\sin (x + π) + 2 \sin (x + π) \cos (x + π)$ .

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Step 1.4.1

Multiply by $1$ .

$\frac{\sin (x + π) \cdot 1 + 2 \sin (x + π) \cos (x + π)}{\cos (x + π)} = 0$

Step 1.4.2

Factor $\sin (x + π)$ out of $2 \sin (x + π) \cos (x + π)$ .

$\frac{\sin (x + π) \cdot 1 + \sin (x + π) (2 \cos (x + π))}{\cos (x + π)} = 0$

Step 1.4.3

Factor $\sin (x + π)$ out of $\sin (x + π) \cdot 1 + \sin (x + π) (2 \cos (x + π))$ .

$\frac{\sin (x + π) (1 + 2 \cos (x + π))}{\cos (x + π)} = 0$

Step 1.5

Separate fractions.

$\frac{1 + 2 \cos (x + π)}{1} \cdot \frac{\sin (x + π)}{\cos (x + π)} = 0$

Step 1.6

Convert from $\frac{\sin (x + π)}{\cos (x + π)}$ to $\tan (x + π)$ .

$\frac{1 + 2 \cos (x + π)}{1} \tan (x + π) = 0$

Step 1.7

Divide $1 + 2 \cos (x + π)$ by $1$ .

$(1 + 2 \cos (x + π)) \tan (x + π) = 0$

Step 1.8

Apply the distributive property.

$1 \tan (x + π) + 2 \cos (x + π) \tan (x + π) = 0$

Step 1.9

Multiply $\tan (x + π)$ by $1$ .

$\tan (x + π) + 2 \cos (x + π) \tan (x + π) = 0$

Step 1.10

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.10.1

Add parentheses.

$\tan (x + π) + 2 (\cos (x + π) \tan (x + π)) = 0$

Step 1.10.2

Reorder $\cos (x + π)$ and $\tan (x + π)$ .

$\tan (x + π) + 2 (\tan (x + π) \cos (x + π)) = 0$

Step 1.10.3

Rewrite $2 \cos (x + π) \tan (x + π)$ in terms of sines and cosines.

$\tan (x + π) + 2 (\frac{\sin (x + π)}{\cos (x + π)} \cos (x + π)) = 0$

Step 1.10.4

Cancel the common factors.

$\tan (x + π) + 2 \sin (x + π) = 0$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 2 π n, \frac{π}{3} + 2 π n, π + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$

Step 3