Trigonometry Examples

$2 \sin (x) \tan (x) = 9 \tan (x)$

Step 1

Subtract $9 \tan (x)$ from both sides of the equation.

$2 \sin (x) \tan (x) - 9 \tan (x) = 0$

Step 2

Factor $\tan (x)$ out of $2 \sin (x) \tan (x) - 9 \tan (x)$ .

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Step 2.1

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

$\tan (x) (2 \sin (x)) - 9 \tan (x) = 0$

Step 2.2

Factor $\tan (x)$ out of $- 9 \tan (x)$ .

$\tan (x) (2 \sin (x)) + \tan (x) \cdot - 9 = 0$

Step 2.3

Factor $\tan (x)$ out of $\tan (x) (2 \sin (x)) + \tan (x) \cdot - 9$ .

$\tan (x) (2 \sin (x) - 9) = 0$

Step 3

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

$\tan (x) = 0$

$2 \sin (x) - 9 = 0$

Step 4

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

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Step 4.1

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

$\tan (x) = 0$

Step 4.2

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

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Step 4.2.1

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

$x = \arctan (0)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

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

$x = 0$

Step 4.2.3

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.

$x = 180 + 0$

Step 4.2.4

Add $180$ and $0$ .

$x = 180$

Step 4.2.5

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

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Step 4.2.5.1

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

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

Step 4.2.5.2

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

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

Step 4.2.5.3

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

$\frac{180}{1}$

Step 4.2.5.4

Divide $180$ by $1$ .

$180$

Step 4.2.6

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 $n$

Step 5

Set $2 \sin (x) - 9$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $2 \sin (x) - 9$ equal to $0$ .

$2 \sin (x) - 9 = 0$

Step 5.2

Solve $2 \sin (x) - 9 = 0$ for $x$ .

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Step 5.2.1

Add $9$ to both sides of the equation.

$2 \sin (x) = 9$

Step 5.2.2

Divide each term in $2 \sin (x) = 9$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 \sin (x) = 9$ by $2$ .

$\frac{2 \sin (x)}{2} = \frac{9}{2}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 \sin (x)}{2} = \frac{9}{2}$

Step 5.2.2.2.1.2

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

$\sin (x) = \frac{9}{2}$

Step 5.2.3

The range of sine is $- 1 \leq y \leq 1$ . Since $\frac{9}{2}$ does not fall in this range, there is no solution.

No solution

Step 6

The final solution is all the values that make $\tan (x) (2 \sin (x) - 9) = 0$ true.

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

Step 7

Consolidate the answers.

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