Solve for x in Degrees tan(x)sin(x)+4sin(x)=0

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Trigonometry

$\tan (x) \sin (x) + 4 \sin (x) = 0$

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.

$\frac{\sin (x)}{\cos (x)} \cdot \sin (x) + 4 \sin (x) = 0$

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

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

$\frac{\sin (x) \sin (x)}{\cos (x)} + 4 \sin (x) = 0$

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

$\frac{\sin (x) \sin (x)}{\cos (x)} + 4 \sin (x) = 0$

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

$\frac{\sin (x) \sin (x)}{\cos (x)} + 4 \sin (x) = 0$

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

$\frac{{\sin (x)}^{1 + 1}}{\cos (x)} + 4 \sin (x) = 0$

Add $1$ and $1$ .

$\frac{\sin^{2} (x)}{\cos (x)} + 4 \sin (x) = 0$

Simplify each term.

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

$\frac{\sin (x) \sin (x)}{\cos (x)} + 4 \sin (x) = 0$

Separate fractions.

$\frac{\sin (x)}{1} \cdot \frac{\sin (x)}{\cos (x)} + 4 \sin (x) = 0$

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

$\frac{\sin (x)}{1} \cdot \tan (x) + 4 \sin (x) = 0$

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

$\sin (x) \tan (x) + 4 \sin (x) = 0$

Factor $\sin (x)$ out of $\sin (x) \tan (x) + 4 \sin (x)$ .

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

$\sin (x) (\tan (x)) + 4 \sin (x) = 0$

Factor $\sin (x)$ out of $4 \sin (x)$ .

$\sin (x) (\tan (x)) + \sin (x) \cdot 4 = 0$

Factor $\sin (x)$ out of $\sin (x) \tan (x) + \sin (x) \cdot 4$ .

$\sin (x) (\tan (x) + 4) = 0$

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

$\sin (x) = 0$

$\tan (x) + 4 = 0$

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 $\tan (x) + 4$ equal to $0$ and solve for $x$ .

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Set $\tan (x) + 4$ equal to $0$ .

$\tan (x) + 4 = 0$

Solve $\tan (x) + 4 = 0$ for $x$ .

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Subtract $4$ from both sides of the equation.

$\tan (x) = - 4$

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

$x = \arctan (- 4)$

Simplify the right side.

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Evaluate $\arctan (- 4)$ .

$x = - 75.96375653$

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.

$x = - 75.96375653 - 180$

Simplify the expression to find the second solution.

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Add $360 °$ to $- 75.96375653 - 180 °$ .

$x = - 75.96375653 - 180 ° + 360 °$

The resulting angle of $104.03624346 °$ is positive and coterminal with $- 75.96375653 - 180$ .

$x = 104.03624346 °$

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 $- 75.96375653$ to find the positive angle.

$- 75.96375653 + 180$

Subtract $75.96375653$ from $180$ .

$104.03624346$

List the new angles.

$x = 104.03624346$

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

$x = 104.03624346 + 180 n, 104.03624346 + 180 n$ , for any integer $n$

The final solution is all the values that make $\sin (x) (\tan (x) + 4) = 0$ true.

$x = 360 n, 180 + 360 n, 104.03624346 + 180 n, 104.03624346 + 180 n$ , for any integer $n$

Consolidate the answers.

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

$x = 180 n, 104.03624346 + 180 n, 104.03624346 + 180 n$ , for any integer $n$

Consolidate $104.03624346 + 180 n$ and $104.03624346 + 180 n$ to $104.03624346 + 180 n$ .

$x = 180 n, 104.03624346 + 180 n$ , for any integer $n$

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