Solve for x in Degrees 3tan(x)sin(x)+2sin(x)=0

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Trigonometry

$3 \tan (x) \sin (x) + 2 \sin (x) = 0$

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.

$3 (\frac{\sin (x)}{\cos (x)}) \cdot \sin (x) + 2 \sin (x) = 0$

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

$\frac{3 \sin (x)}{\cos (x)} \cdot \sin (x) + 2 \sin (x) = 0$

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

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

$\frac{3 \sin (x) \sin (x)}{\cos (x)} + 2 \sin (x) = 0$

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

$\frac{3 (\sin (x) \sin (x))}{\cos (x)} + 2 \sin (x) = 0$

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

$\frac{3 (\sin (x) \sin (x))}{\cos (x)} + 2 \sin (x) = 0$

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

$\frac{3 {\sin (x)}^{1 + 1}}{\cos (x)} + 2 \sin (x) = 0$

Add $1$ and $1$ .

$\frac{3 \sin^{2} (x)}{\cos (x)} + 2 \sin (x) = 0$

Simplify each term.

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

$\frac{3 (\sin (x) \sin (x))}{\cos (x)} + 2 \sin (x) = 0$

Separate fractions.

$\frac{3 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + 2 \sin (x) = 0$

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

$\frac{3 (\sin (x))}{1} \cdot \tan (x) + 2 \sin (x) = 0$

Divide $3 (\sin (x))$ by $1$ .

$3 \sin (x) \tan (x) + 2 \sin (x) = 0$

Factor $\sin (x)$ out of $3 \sin (x) \tan (x) + 2 \sin (x)$ .

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Factor $\sin (x)$ out of $3 \sin (x) \tan (x)$ .

$\sin (x) (3 \tan (x)) + 2 \sin (x) = 0$

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

$\sin (x) (3 \tan (x)) + \sin (x) \cdot 2 = 0$

Factor $\sin (x)$ out of $\sin (x) (3 \tan (x)) + \sin (x) \cdot 2$ .

$\sin (x) (3 \tan (x) + 2) = 0$

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

$\sin (x) = 0$

$3 \tan (x) + 2 = 0$

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

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

$\sin (x) = 0$

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

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

$x = \arcsin (0)$

Simplify the right side.

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

$x = 0$

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

$x = 180 - 0$

Subtract $0$ from $180$ .

$x = 180$

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 $1$ .

$360$

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

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

Set $3 \tan (x) + 2$ equal to $0$ and solve for $x$ .

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Set $3 \tan (x) + 2$ equal to $0$ .

$3 \tan (x) + 2 = 0$

Solve $3 \tan (x) + 2 = 0$ for $x$ .

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Subtract $2$ from both sides of the equation.

$3 \tan (x) = - 2$

Divide each term in $3 \tan (x) = - 2$ by $3$ and simplify.

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Divide each term in $3 \tan (x) = - 2$ by $3$ .

$\frac{3 \tan (x)}{3} = \frac{- 2}{3}$

Simplify the left side.

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Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 \tan (x)}{3} = \frac{- 2}{3}$

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

$\tan (x) = \frac{- 2}{3}$

Simplify the right side.

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Move the negative in front of the fraction.

$\tan (x) = - \frac{2}{3}$

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

$x = \arctan (- \frac{2}{3})$

Simplify the right side.

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Evaluate $\arctan (- \frac{2}{3})$ .

$x = - 33.69006752$

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

$x = - 33.69006752 - 180$

Simplify the expression to find the second solution.

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Add $360 °$ to $- 33.69006752 - 180 °$ .

$x = - 33.69006752 - 180 ° + 360 °$

The resulting angle of $146.30993247 °$ is positive and coterminal with $- 33.69006752 - 180$ .

$x = 146.30993247 °$

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 $1$ .

$180$

Add $180$ to every negative angle to get positive angles.

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Add $180$ to $- 33.69006752$ to find the positive angle.

$- 33.69006752 + 180$

Subtract $33.69006752$ from $180$ .

$146.30993247$

List the new angles.

$x = 146.30993247$

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

$x = 146.30993247 + 180 n, 146.30993247 + 180 n$ , for any integer $n$

The final solution is all the values that make $\sin (x) (3 \tan (x) + 2) = 0$ true.

$x = 360 n, 180 + 360 n, 146.30993247 + 180 n, 146.30993247 + 180 n$ , for any integer $n$

Consolidate the answers.

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Consolidate $360 n$ and $180 + 360 n$ to $180 n$ .

$x = 180 n, 146.30993247 + 180 n, 146.30993247 + 180 n$ , for any integer $n$

Consolidate $146.30993247 + 180 n$ and $146.30993247 + 180 n$ to $146.30993247 + 180 n$ .

$x = 180 n, 146.30993247 + 180 n$ , for any integer $n$

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