Solve for x tan(x)sin(x)^2=3tan(x)

$\tan (x) \sin^{2} (x) = 3 \tan (x)$

Step 1

Subtract $3 \tan (x)$ from both sides of the equation.

$\tan (x) \sin^{2} (x) - 3 \tan (x) = 0$

Step 2

Simplify the left side of the equation.

Simplify each term.

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

$\frac{\sin (x)}{\cos (x)} \cdot \sin^{2} (x) - 3 \tan (x) = 0$

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

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

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

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

Multiply $\sin (x)$ by $\sin^{2} (x)$ .

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

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

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

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

Add $1$ and $2$ .

$\frac{\sin^{3} (x)}{\cos (x)} - 3 \tan (x) = 0$

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

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

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

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

Move the negative in front of the fraction.

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

Simplify each term.

Factor $\sin (x)$ out of $\sin^{3} (x)$ .

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

Separate fractions.

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

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

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

Divide $\sin^{2} (x)$ by $1$ .

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

Separate fractions.

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

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

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

Divide $3$ by $1$ .

$\sin^{2} (x) \tan (x) - (3 \tan (x)) = 0$

Multiply $3$ by $- 1$ .

$\sin^{2} (x) \tan (x) - 3 \tan (x) = 0$

Step 3

Factor $\tan (x)$ out of $\sin^{2} (x) \tan (x) - 3 \tan (x)$ .

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

$\tan (x) \sin^{2} (x) - 3 \tan (x) = 0$

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

$\tan (x) \sin^{2} (x) + \tan (x) \cdot - 3 = 0$

Factor $\tan (x)$ out of $\tan (x) \sin^{2} (x) + \tan (x) \cdot - 3$ .

$\tan (x) (\sin^{2} (x) - 3) = 0$

Step 4

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

$\tan (x) = 0$

$\sin^{2} (x) - 3 = 0$

Step 5

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

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

$\tan (x) = 0$

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

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

$x = \arctan (0)$

Simplify the right side.

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, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + 0$

Add $π$ and $0$ .

$x = π$

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

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

$\frac{π}{| b |}$

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

$\frac{π}{| 1 |}$

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

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

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

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

Step 6

Set $\sin^{2} (x) - 3$ equal to $0$ and solve for $x$ .

Set $\sin^{2} (x) - 3$ equal to $0$ .

$\sin^{2} (x) - 3 = 0$

Solve $\sin^{2} (x) - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$\sin^{2} (x) = 3$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$\sin (x) = \pm \sqrt{3}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$\sin (x) = \sqrt{3}$

Next, use the negative value of the $\pm$ to find the second solution.

$\sin (x) = - \sqrt{3}$

The complete solution is the result of both the positive and negative portions of the solution.

$\sin (x) = \sqrt{3}, - \sqrt{3}$

Set up each of the solutions to solve for $x$ .

$\sin (x) = \sqrt{3}$

$\sin (x) = - \sqrt{3}$

Solve for $x$ in $\sin (x) = \sqrt{3}$ .

The range of sine is $- 1 \leq y \leq 1$ . Since $\sqrt{3}$ does not fall in this range, there is no solution.

No solution

Solve for $x$ in $\sin (x) = - \sqrt{3}$ .

The range of sine is $- 1 \leq y \leq 1$ . Since $- \sqrt{3}$ does not fall in this range, there is no solution.

No solution

Step 7

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

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

Step 8

Consolidate the answers.

$x = π n$ , for any integer $n$

Solve for x tan(x)sin(x)^2=3tan(x)

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