Trigonometry Examples

$5 \tan (x) \sin (x) - 4 \sin (x) = 0$

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.

$5 (\frac{\sin (x)}{\cos (x)}) \cdot \sin (x) - 4 \sin (x) = 0$

Step 1.1.2

Combine

$5$ and

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

$\frac{5 \sin (x)}{\cos (x)} \cdot \sin (x) - 4 \sin (x) = 0$

Step 1.1.3

Multiply

$\frac{5 \sin (x)}{\cos (x)} \sin (x)$ .

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Step 1.1.3.1

Combine

$\frac{5 \sin (x)}{\cos (x)}$ and

$\sin (x)$ .

$\frac{5 \sin (x) \sin (x)}{\cos (x)} - 4 \sin (x) = 0$

Step 1.1.3.2

Raise

$\sin (x)$ to the power of

$1$ .

$\frac{5 (\sin (x) \sin (x))}{\cos (x)} - 4 \sin (x) = 0$

Step 1.1.3.3

Raise

$\sin (x)$ to the power of

$1$ .

$\frac{5 (\sin (x) \sin (x))}{\cos (x)} - 4 \sin (x) = 0$

Step 1.1.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5 {\sin (x)}^{1 + 1}}{\cos (x)} - 4 \sin (x) = 0$

Step 1.1.3.5

Add

$1$ and

$1$ .

$\frac{5 \sin^{2} (x)}{\cos (x)} - 4 \sin (x) = 0$

Step 1.2

Simplify each term.

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Step 1.2.1

Factor

$\sin (x)$ out of

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

$\frac{5 (\sin (x) \sin (x))}{\cos (x)} - 4 \sin (x) = 0$

Step 1.2.2

Separate fractions.

$\frac{5 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} - 4 \sin (x) = 0$

Step 1.2.3

Convert from

$\frac{\sin (x)}{\cos (x)}$ to

$\tan (x)$ .

$\frac{5 (\sin (x))}{1} \cdot \tan (x) - 4 \sin (x) = 0$

Step 1.2.4

Divide

$5 (\sin (x))$ by

$1$ .

$5 \sin (x) \tan (x) - 4 \sin (x) = 0$

Step 2

Factor

$\sin (x)$ out of

$5 \sin (x) \tan (x) - 4 \sin (x)$ .

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Step 2.1

Factor

$\sin (x)$ out of

$5 \sin (x) \tan (x)$ .

$\sin (x) (5 \tan (x)) - 4 \sin (x) = 0$

Step 2.2

Factor

$\sin (x)$ out of

$- 4 \sin (x)$ .

$\sin (x) (5 \tan (x)) + \sin (x) \cdot - 4 = 0$

Step 2.3

Factor

$\sin (x)$ out of

$\sin (x) (5 \tan (x)) + \sin (x) \cdot - 4$ .

$\sin (x) (5 \tan (x) - 4) = 0$

Step 3

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$\sin (x) = 0$

$5 \tan (x) - 4 = 0$

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

$n$

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

$n$

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

$n$

Step 5

Set

$5 \tan (x) - 4$ equal to

$0$ and solve for

$x$ .

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Step 5.1

Set

$5 \tan (x) - 4$ equal to

$0$ .

$5 \tan (x) - 4 = 0$

Step 5.2

Solve

$5 \tan (x) - 4 = 0$ for

$x$ .

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Step 5.2.1

Add

$4$ to both sides of the equation.

$5 \tan (x) = 4$

Step 5.2.2

Divide each term in

$5 \tan (x) = 4$ by

$5$ and simplify.

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Step 5.2.2.1

Divide each term in

$5 \tan (x) = 4$ by

$5$ .

$\frac{5 \tan (x)}{5} = \frac{4}{5}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of

$5$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{5 \tan (x)}{5} = \frac{4}{5}$

Step 5.2.2.2.1.2

Divide

$\tan (x)$ by

$1$ .

$\tan (x) = \frac{4}{5}$

Step 5.2.3

Take the inverse tangent of both sides of the equation to extract

$x$ from inside the tangent.

$x = \arctan (\frac{4}{5})$

Step 5.2.4

Simplify the right side.

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Step 5.2.4.1

Evaluate

$\arctan (\frac{4}{5})$ .

$x = 38.65980825$

Step 5.2.5

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.

$x = 180 + 38.65980825$

Step 5.2.6

Add

$180$ and

$38.65980825$ .

$x = 218.65980825$

Step 5.2.7

Find the period of

$\tan (x)$ .

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Step 5.2.7.1

The period of the function can be calculated using

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

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

Step 5.2.7.2

Replace

$b$ with

$1$ in the formula for period.

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

Step 5.2.7.3

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

$0$ and

$1$ is

$1$ .

$\frac{180}{1}$

Step 5.2.7.4

Divide

$180$ by

$1$ .

$180$

Step 5.2.8

The period of the

$\tan (x)$ function is

$180$ so values will repeat every

$180$ degrees in both directions.

$x = 38.65980825 + 180 n, 218.65980825 + 180 n$ , for any integer

$n$

$x = 38.65980825 + 180 n, 218.65980825 + 180 n$ , for any integer

$n$

$x = 38.65980825 + 180 n, 218.65980825 + 180 n$ , for any integer

$n$

Step 6

The final solution is all the values that make

$\sin (x) (5 \tan (x) - 4) = 0$ true.

$x = 360 n, 180 + 360 n, 38.65980825 + 180 n, 218.65980825 + 180 n$ , for any integer

$n$

Step 7

Consolidate the answers.

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Step 7.1

Consolidate

$360 n$ and

$180 + 360 n$ to

$180 n$ .

$x = 180 n, 38.65980825 + 180 n, 218.65980825 + 180 n$ , for any integer

$n$

Step 7.2

Consolidate

$38.65980825 + 180 n$ and

$218.65980825 + 180 n$ to

$38.65980825 + 180 n$ .

$x = 180 n, 38.65980825 + 180 n$ , for any integer

$n$

$x = 180 n, 38.65980825 + 180 n$ , for any integer

$n$