Trigonometry Examples

$8 \sin (x) \tan (x) - 7 \tan (x) = 0$

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.

$8 \sin (x) (\frac{\sin (x)}{\cos (x)}) - 7 \tan (x) = 0$

Step 1.1.2

Multiply $8 \sin (x) \frac{\sin (x)}{\cos (x)}$ .

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Step 1.1.2.1

Combine $\frac{\sin (x)}{\cos (x)}$ and $8$ .

$\frac{\sin (x) \cdot 8}{\cos (x)} \cdot \sin (x) - 7 \tan (x) = 0$

Step 1.1.2.2

Combine $\frac{\sin (x) \cdot 8}{\cos (x)}$ and $\sin (x)$ .

$\frac{\sin (x) \cdot (8 \sin (x))}{\cos (x)} - 7 \tan (x) = 0$

Step 1.1.2.3

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

$\frac{8 (\sin (x) \sin (x))}{\cos (x)} - 7 \tan (x) = 0$

Step 1.1.2.4

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

$\frac{8 (\sin (x) \sin (x))}{\cos (x)} - 7 \tan (x) = 0$

Step 1.1.2.5

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

$\frac{8 {\sin (x)}^{1 + 1}}{\cos (x)} - 7 \tan (x) = 0$

Step 1.1.2.6

Add $1$ and $1$ .

$\frac{8 \sin^{2} (x)}{\cos (x)} - 7 \tan (x) = 0$

Step 1.1.3

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

$\frac{8 \sin^{2} (x)}{\cos (x)} - 7 \frac{\sin (x)}{\cos (x)} = 0$

Step 1.1.4

Combine $- 7$ and $\frac{\sin (x)}{\cos (x)}$ .

$\frac{8 \sin^{2} (x)}{\cos (x)} + \frac{- 7 \sin (x)}{\cos (x)} = 0$

Step 1.1.5

Move the negative in front of the fraction.

$\frac{8 \sin^{2} (x)}{\cos (x)} - \frac{7 \sin (x)}{\cos (x)} = 0$

Step 1.2

Simplify each term.

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Step 1.2.1

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

$\frac{8 (\sin (x) \sin (x))}{\cos (x)} - \frac{7 \sin (x)}{\cos (x)} = 0$

Step 1.2.2

Separate fractions.

$\frac{8 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} - \frac{7 \sin (x)}{\cos (x)} = 0$

Step 1.2.3

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

$\frac{8 (\sin (x))}{1} \cdot \tan (x) - \frac{7 \sin (x)}{\cos (x)} = 0$

Step 1.2.4

Divide $8 (\sin (x))$ by $1$ .

$8 (\sin (x)) \tan (x) - \frac{7 \sin (x)}{\cos (x)} = 0$

Step 1.2.5

Separate fractions.

$8 \sin (x) \tan (x) - (\frac{7}{1} \cdot \frac{\sin (x)}{\cos (x)}) = 0$

Step 1.2.6

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

$8 \sin (x) \tan (x) - (\frac{7}{1} \cdot \tan (x)) = 0$

Step 1.2.7

Divide $7$ by $1$ .

$8 \sin (x) \tan (x) - (7 \tan (x)) = 0$

Step 1.2.8

Multiply $7$ by $- 1$ .

$8 \sin (x) \tan (x) - 7 \tan (x) = 0$

Step 2

Factor $\tan (x)$ out of $8 \sin (x) \tan (x) - 7 \tan (x)$ .

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Step 2.1

Factor $\tan (x)$ out of $8 \sin (x) \tan (x)$ .

$\tan (x) (8 \sin (x)) - 7 \tan (x) = 0$

Step 2.2

Factor $\tan (x)$ out of $- 7 \tan (x)$ .

$\tan (x) (8 \sin (x)) + \tan (x) \cdot - 7 = 0$

Step 2.3

Factor $\tan (x)$ out of $\tan (x) (8 \sin (x)) + \tan (x) \cdot - 7$ .

$\tan (x) (8 \sin (x) - 7) = 0$

Step 3

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

$\tan (x) = 0$

$8 \sin (x) - 7 = 0$

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 $8 \sin (x) - 7$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $8 \sin (x) - 7$ equal to $0$ .

$8 \sin (x) - 7 = 0$

Step 5.2

Solve $8 \sin (x) - 7 = 0$ for $x$ .

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Step 5.2.1

Add $7$ to both sides of the equation.

$8 \sin (x) = 7$

Step 5.2.2

Divide each term in $8 \sin (x) = 7$ by $8$ and simplify.

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Step 5.2.2.1

Divide each term in $8 \sin (x) = 7$ by $8$ .

$\frac{8 \sin (x)}{8} = \frac{7}{8}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $8$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{8 \sin (x)}{8} = \frac{7}{8}$

Step 5.2.2.2.1.2

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

$\sin (x) = \frac{7}{8}$

Step 5.2.3

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

$x = \arcsin (\frac{7}{8})$

Step 5.2.4

Simplify the right side.

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Step 5.2.4.1

Evaluate $\arcsin (\frac{7}{8})$ .

$x = 61.04497562$

Step 5.2.5

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.

$x = 180 - 61.04497562$

Step 5.2.6

Subtract $61.04497562$ from $180$ .

$x = 118.95502437$

Step 5.2.7

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

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Step 5.2.7.1

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

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

Step 5.2.7.2

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

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

Step 5.2.7.3

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

$\frac{360}{1}$

Step 5.2.7.4

Divide $360$ by $1$ .

$360$

Step 5.2.8

The period of the $\sin (x)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$x = 61.04497562 + 360 n, 118.95502437 + 360 n$ , for any integer $n$

Step 6

The final solution is all the values that make $\tan (x) (8 \sin (x) - 7) = 0$ true.

$x = 180 n, 180 + 180 n, 61.04497562 + 360 n, 118.95502437 + 360 n$ , for any integer $n$

Step 7

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

$x = 180 n, 61.04497562 + 360 n, 118.95502437 + 360 n$ , for any integer $n$