Trigonometry Examples

$3 \tan (x) \sin (x) = \sin (x)$

Step 1

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

$3 \tan (x) \sin (x) - \sin (x) = 0$

Step 2

Simplify the left side of the equation.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

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

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

Step 2.1.3.5

Add $1$ and $1$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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

Step 2.2.2

Separate fractions.

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

Step 2.2.3

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

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

Step 2.2.4

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

$3 \sin (x) \tan (x) - \sin (x) = 0$

Step 3

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

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Step 3.1

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

$\sin (x) (3 \tan (x)) - \sin (x) = 0$

Step 3.2

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

$\sin (x) (3 \tan (x)) + \sin (x) \cdot - 1 = 0$

Step 3.3

Factor $\sin (x)$ out of $\sin (x) (3 \tan (x)) + \sin (x) \cdot - 1$ .

$\sin (x) (3 \tan (x) - 1) = 0$

Step 4

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

$\sin (x) = 0$

$3 \tan (x) - 1 = 0$

Step 5

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

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Step 5.1

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

$\sin (x) = 0$

Step 5.2

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

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Step 5.2.1

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

$x = \arcsin (0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

$x = 0$

Step 5.2.3

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 5.2.4

Subtract $0$ from $180$ .

$x = 180$

Step 5.2.5

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

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Step 5.2.5.1

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

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

Step 5.2.5.2

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

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

Step 5.2.5.3

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

$\frac{360}{1}$

Step 5.2.5.4

Divide $360$ by $1$ .

$360$

Step 5.2.6

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 6

Set $3 \tan (x) - 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $3 \tan (x) - 1$ equal to $0$ .

$3 \tan (x) - 1 = 0$

Step 6.2

Solve $3 \tan (x) - 1 = 0$ for $x$ .

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Step 6.2.1

Add $1$ to both sides of the equation.

$3 \tan (x) = 1$

Step 6.2.2

Divide each term in $3 \tan (x) = 1$ by $3$ and simplify.

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Step 6.2.2.1

Divide each term in $3 \tan (x) = 1$ by $3$ .

$\frac{3 \tan (x)}{3} = \frac{1}{3}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{3 \tan (x)}{3} = \frac{1}{3}$

Step 6.2.2.2.1.2

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

$\tan (x) = \frac{1}{3}$

Step 6.2.3

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

$x = \arctan (\frac{1}{3})$

Step 6.2.4

Simplify the right side.

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Step 6.2.4.1

Evaluate $\arctan (\frac{1}{3})$ .

$x = 18.43494882$

Step 6.2.5

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.

$x = 180 + 18.43494882$

Step 6.2.6

Add $180$ and $18.43494882$ .

$x = 198.43494882$

Step 6.2.7

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

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Step 6.2.7.1

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

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

Step 6.2.7.2

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

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

Step 6.2.7.3

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

$\frac{180}{1}$

Step 6.2.7.4

Divide $180$ by $1$ .

$180$

Step 6.2.8

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

$x = 18.43494882 + 180 n, 198.43494882 + 180 n$ , for any integer $n$

Step 7

The final solution is all the values that make $\sin (x) (3 \tan (x) - 1) = 0$ true.

$x = 360 n, 180 + 360 n, 18.43494882 + 180 n, 198.43494882 + 180 n$ , for any integer $n$

Step 8

Consolidate the answers.

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Step 8.1

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

$x = 180 n, 18.43494882 + 180 n, 198.43494882 + 180 n$ , for any integer $n$

Step 8.2

Consolidate $18.43494882 + 180 n$ and $198.43494882 + 180 n$ to $18.43494882 + 180 n$ .

$x = 180 n, 18.43494882 + 180 n$ , for any integer $n$