Trigonometry Examples

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

Step 1

Simplify the left side of the equation.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Add $1$ and $1$ .

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

Step 1.2

Simplify each term.

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Step 1.2.1

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

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

Step 1.2.2

Separate fractions.

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

$\sin (x) = 0$

$\tan (x) + 4 = 0$

Step 4

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

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Step 4.1

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

$\sin (x) = 0$

Step 4.2

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

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Step 4.2.1

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

$x = \arcsin (0)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

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

$x = 0$

Step 4.2.3

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.

$x = 180 - 0$

Step 4.2.4

Subtract $0$ from $180$ .

$x = 180$

Step 4.2.5

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

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Step 4.2.5.1

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

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

Step 4.2.5.2

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

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

Step 4.2.5.3

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

$\frac{360}{1}$

Step 4.2.5.4

Divide $360$ by $1$ .

$360$

Step 4.2.6

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 $n$

Step 5

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

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Step 5.1

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

$\tan (x) + 4 = 0$

Step 5.2

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

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Step 5.2.1

Subtract $4$ from both sides of the equation.

$\tan (x) = - 4$

Step 5.2.2

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

$x = \arctan (- 4)$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Evaluate $\arctan (- 4)$ .

$x = - 75.96375653$

Step 5.2.4

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.

$x = - 75.96375653 - 180$

Step 5.2.5

Simplify the expression to find the second solution.

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Step 5.2.5.1

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

$x = - 75.96375653 - 180 ° + 360 °$

Step 5.2.5.2

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

$x = 104.03624346 °$

Step 5.2.6

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

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Step 5.2.6.1

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

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

Step 5.2.6.2

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

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

Step 5.2.6.3

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

$\frac{180}{1}$

Step 5.2.6.4

Divide $180$ by $1$ .

$180$

Step 5.2.7

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

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Step 5.2.7.1

Add $180$ to $- 75.96375653$ to find the positive angle.

$- 75.96375653 + 180$

Step 5.2.7.2

Subtract $75.96375653$ from $180$ .

$104.03624346$

Step 5.2.7.3

List the new angles.

$x = 104.03624346$

Step 5.2.8

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 $n$

Step 6

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 $n$

Step 7

Consolidate the answers.

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Step 7.1

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

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

Step 7.2

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 $n$