Solve for x in Degrees 2sin(x)tan(x)+tan(x)=0

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Trigonometry

$2 \sin (x) \tan (x) + \tan (x) = 0$

Simplify the left side of the equation.

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Simplify each term.

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

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

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

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

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

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

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

Raise $\sin (x)$ to the power of $1$ .

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

Raise $\sin (x)$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

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

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

Simplify each term.

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Factor $\sin (x)$ out of $\sin^{2} (x)$ .

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

Separate fractions.

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

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

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

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

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

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

$2 \sin (x) \tan (x) + \tan (x) = 0$

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

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Factor $\tan (x)$ out of $2 \sin (x) \tan (x)$ .

$\tan (x) (2 \sin (x)) + \tan (x) = 0$

Raise $\tan (x)$ to the power of $1$ .

$\tan (x) (2 \sin (x)) + \tan (x) = 0$

Factor $\tan (x)$ out of $\tan^{1} (x)$ .

$\tan (x) (2 \sin (x)) + \tan (x) \cdot 1 = 0$

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

$\tan (x) (2 \sin (x) + 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\tan (x) = 0$

$2 \sin (x) + 1 = 0$

Set $\tan (x)$ equal to $0$ and solve for $x$ .

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Set $\tan (x)$ equal to $0$ .

$\tan (x) = 0$

Solve $\tan (x) = 0$ for $x$ .

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Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$x = \arctan (0)$

Simplify the right side.

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The exact value of $\arctan (0)$ is $0$ .

$x = 0$

The tangent function is positive in the first and third quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the fourth quadrant.

$x = 180 + 0$

Add $180$ and $0$ .

$x = 180$

Find the period of $\tan (x)$ .

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The period of the function can be calculated using $\frac{180}{| b |}$ .

$\frac{180}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{180}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{180}{1}$

Divide $180$ by $1$ .

$180$

The period of the $\tan (x)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$x = 180 n, 180 + 180 n$ , for any integer $n$

Set $2 \sin (x) + 1$ equal to $0$ and solve for $x$ .

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Set $2 \sin (x) + 1$ equal to $0$ .

$2 \sin (x) + 1 = 0$

Solve $2 \sin (x) + 1 = 0$ for $x$ .

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Subtract $1$ from both sides of the equation.

$2 \sin (x) = - 1$

Divide each term in $2 \sin (x) = - 1$ by $2$ and simplify.

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Divide each term in $2 \sin (x) = - 1$ by $2$ .

$\frac{2 \sin (x)}{2} = \frac{- 1}{2}$

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 \sin (x)}{2} = \frac{- 1}{2}$

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

$\sin (x) = \frac{- 1}{2}$

Simplify the right side.

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Move the negative in front of the fraction.

$\sin (x) = - \frac{1}{2}$

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (- \frac{1}{2})$

Simplify the right side.

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

$x = - 30$

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $360$ , to find a reference angle. Next, add this reference angle to $180$ to find the solution in the third quadrant.

$x = 360 + 30 + 180$

Simplify the expression to find the second solution.

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Subtract $360 °$ from $360 + 30 + 180 °$ .

$x = 360 + 30 + 180 ° - 360 °$

The resulting angle of $210 °$ is positive, less than $360 °$ , and coterminal with $360 + 30 + 180$ .

$x = 210 °$

Find the period of $\sin (x)$ .

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The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Divide $360$ by $1$ .

$360$

Add $360$ to every negative angle to get positive angles.

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Add $360$ to $- 30$ to find the positive angle.

$- 30 + 360$

Subtract $30$ from $360$ .

$330$

List the new angles.

$x = 330$

The period of the $\sin (x)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$x = 210 + 360 n, 330 + 360 n$ , for any integer $n$

The final solution is all the values that make $\tan (x) (2 \sin (x) + 1) = 0$ true.

$x = 180 n, 180 + 180 n, 210 + 360 n, 330 + 360 n$ , for any integer $n$

Consolidate $180 n$ and $180 + 180 n$ to $180 n$ .

$x = 180 n, 210 + 360 n, 330 + 360 n$ , for any integer $n$

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