Trigonometry Examples

2 sin (x) tan (x) + tan (x) = 0

$2 \sin (x) \tan (x) + \tan (x) = 0$

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)

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

2 sin (x) (\frac{sin (x)}{cos (x)}) + tan (x) = 0

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

Step 1.1.2

Multiply

2 sin (x) \frac{sin (x)}{cos (x)}

$2 \sin (x) \frac{\sin (x)}{\cos (x)}$ .

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Step 1.1.2.1

Combine

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

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

2

$2$ .

\frac{sin (x) \cdot 2}{cos (x)} \cdot sin (x) + tan (x) = 0

$\frac{\sin (x) \cdot 2}{\cos (x)} \cdot \sin (x) + \tan (x) = 0$

Step 1.1.2.2

Combine

\frac{sin (x) \cdot 2}{cos (x)}

$\frac{\sin (x) \cdot 2}{\cos (x)}$ and

sin (x)

$\sin (x)$ .

\frac{sin (x) \cdot (2 sin (x))}{cos (x)} + tan (x) = 0

$\frac{\sin (x) \cdot (2 \sin (x))}{\cos (x)} + \tan (x) = 0$

Step 1.1.2.3

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

\frac{2 (sin (x) sin (x))}{cos (x)} + tan (x) = 0

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

Step 1.1.2.4

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

\frac{2 (sin (x) sin (x))}{cos (x)} + tan (x) = 0

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

Step 1.1.2.5

Use the power rule

a^{m} a^{n} = a^{m + n}

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

\frac{2 {sin (x)}^{1 + 1}}{cos (x)} + tan (x) = 0

$\frac{2 {\sin (x)}^{1 + 1}}{\cos (x)} + \tan (x) = 0$

Step 1.1.2.6

Add

1

$1$ and

1

$1$ .

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

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

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

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

Step 1.1.3

Rewrite

tan (x)

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

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

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

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

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

Step 1.2

Simplify each term.

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Step 1.2.1

Factor

sin (x)

$\sin (x)$ out of

{sin}^{2} (x)

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

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

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

Step 1.2.2

Separate fractions.

\frac{2 (sin (x))}{1} \cdot \frac{sin (x)}{cos (x)} + \frac{sin (x)}{cos (x)} = 0

$\frac{2 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} = 0$

Step 1.2.3

Convert from

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

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

tan (x)

$\tan (x)$ .

\frac{2 (sin (x))}{1} \cdot tan (x) + \frac{sin (x)}{cos (x)} = 0

$\frac{2 (\sin (x))}{1} \cdot \tan (x) + \frac{\sin (x)}{\cos (x)} = 0$

Step 1.2.4

Divide

2 (sin (x))

$2 (\sin (x))$ by

1

$1$ .

2 (sin (x)) tan (x) + \frac{sin (x)}{cos (x)} = 0

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

Step 1.2.5

Convert from

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

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

tan (x)

$\tan (x)$ .

2 sin (x) tan (x) + tan (x) = 0

$2 \sin (x) \tan (x) + \tan (x) = 0$

2 sin (x) tan (x) + tan (x) = 0

$2 \sin (x) \tan (x) + \tan (x) = 0$

2 sin (x) tan (x) + tan (x) = 0

$2 \sin (x) \tan (x) + \tan (x) = 0$

Step 2

Factor

tan (x)

$\tan (x)$ out of

2 sin (x) tan (x) + tan (x)

$2 \sin (x) \tan (x) + \tan (x)$ .

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Step 2.1

Factor

tan (x)

$\tan (x)$ out of

2 sin (x) tan (x)

$2 \sin (x) \tan (x)$ .

tan (x) (2 sin (x)) + tan (x) = 0

$\tan (x) (2 \sin (x)) + \tan (x) = 0$

Step 2.2

Raise

tan (x)

$\tan (x)$ to the power of

1

$1$ .

tan (x) (2 sin (x)) + tan (x) = 0

$\tan (x) (2 \sin (x)) + \tan (x) = 0$

Step 2.3

Factor

tan (x)

$\tan (x)$ out of

{tan}^{1} (x)

$\tan^{1} (x)$ .

tan (x) (2 sin (x)) + tan (x) \cdot 1 = 0

$\tan (x) (2 \sin (x)) + \tan (x) \cdot 1 = 0$

Step 2.4

Factor

tan (x)

$\tan (x)$ out of

tan (x) (2 sin (x)) + tan (x) \cdot 1

$\tan (x) (2 \sin (x)) + \tan (x) \cdot 1$ .

tan (x) (2 sin (x) + 1) = 0

$\tan (x) (2 \sin (x) + 1) = 0$

tan (x) (2 sin (x) + 1) = 0

$\tan (x) (2 \sin (x) + 1) = 0$

Step 3

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

0

$0$ , the entire expression will be equal to

0

$0$ .

tan (x) = 0

$\tan (x) = 0$

2 sin (x) + 1 = 0

$2 \sin (x) + 1 = 0$

Step 4

Set

tan (x)

$\tan (x)$ equal to

0

$0$ and solve for

x

$x$ .

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Step 4.1

Set

tan (x)

$\tan (x)$ equal to

0

$0$ .

tan (x) = 0

$\tan (x) = 0$

Step 4.2

Solve

tan (x) = 0

$\tan (x) = 0$ for

x

$x$ .

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Step 4.2.1

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

x

$x$ from inside the tangent.

x = arctan (0)

$x = \arctan (0)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

The exact value of

arctan (0)

$\arctan (0)$ is

0

$0$ .

x = 0

$x = 0$

x = 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

$180$ to find the solution in the fourth quadrant.

x = 180 + 0

$x = 180 + 0$

Step 4.2.4

Add

180

$180$ and

0

$0$ .

x = 180

$x = 180$

Step 4.2.5

Find the period of

tan (x)

$\tan (x)$ .

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Step 4.2.5.1

The period of the function can be calculated using

\frac{180}{| b |}

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

\frac{180}{| b |}

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

Step 4.2.5.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{180}{| 1 |}

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

Step 4.2.5.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{180}{1}

$\frac{180}{1}$

Step 4.2.5.4

Divide

180

$180$ by

1

$1$ .

180

$180$

180

$180$

Step 4.2.6

The period of the

tan (x)

$\tan (x)$ function is

180

$180$ so values will repeat every

180

$180$ degrees in both directions.

x = 180 n, 180 + 180 n

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

n

$n$

x = 180 n, 180 + 180 n

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

n

$n$

x = 180 n, 180 + 180 n

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

n

$n$

Step 5

Set

2 sin (x) + 1

$2 \sin (x) + 1$ equal to

0

$0$ and solve for

x

$x$ .

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Step 5.1

Set

2 sin (x) + 1

$2 \sin (x) + 1$ equal to

0

$0$ .

2 sin (x) + 1 = 0

$2 \sin (x) + 1 = 0$

Step 5.2

Solve

2 sin (x) + 1 = 0

$2 \sin (x) + 1 = 0$ for

x

$x$ .

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Step 5.2.1

Subtract

1

$1$ from both sides of the equation.

2 sin (x) = - 1

$2 \sin (x) = - 1$

Step 5.2.2

Divide each term in

2 sin (x) = - 1

$2 \sin (x) = - 1$ by

2

$2$ and simplify.

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Step 5.2.2.1

Divide each term in

2 sin (x) = - 1

$2 \sin (x) = - 1$ by

2

$2$ .

\frac{2 sin (x)}{2} = \frac{- 1}{2}

$\frac{2 \sin (x)}{2} = \frac{- 1}{2}$

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

$2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 \sin (x)}{2} = \frac{- 1}{2}$

Step 5.2.2.2.1.2

Divide

$\sin (x)$ by

$1$ .

$\sin (x) = \frac{- 1}{2}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

$\sin (x) = - \frac{1}{2}$

Step 5.2.3

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

$x$ from inside the sine.

$x = \arcsin (- \frac{1}{2})$

Step 5.2.4

Simplify the right side.

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Step 5.2.4.1

The exact value of

$\arcsin (- \frac{1}{2})$ is

$- 30$ .

$x = - 30$

Step 5.2.5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from

$360$ , to find a reference angle. Next, add this reference angle to

$180$ to find the solution in the third quadrant.

$x = 360 + 30 + 180$

Step 5.2.6

Simplify the expression to find the second solution.

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Step 5.2.6.1

Subtract

$360 °$ from

$360 + 30 + 180 °$ .

$x = 360 + 30 + 180 ° - 360 °$

Step 5.2.6.2

The resulting angle of

$210 °$ is positive, less than

$360 °$ , and coterminal with

$360 + 30 + 180$ .

$x = 210 °$

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

Add

$360$ to every negative angle to get positive angles.

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Step 5.2.8.1

Add

$360$ to

$- 30$ to find the positive angle.

$- 30 + 360$

Step 5.2.8.2

Subtract

$30$ from

$360$ .

$330$

Step 5.2.8.3

List the new angles.

$x = 330$

Step 5.2.9

The period of the

$\sin (x)$ function is

$360$ so values will repeat every

$360$ degrees in both directions.

$x = 210 + 360 n, 330 + 360 n$ , for any integer

$n$

$x = 210 + 360 n, 330 + 360 n$ , for any integer

$n$

$x = 210 + 360 n, 330 + 360 n$ , for any integer

$n$

Step 6

The final solution is all the values that make

$\tan (x) (2 \sin (x) + 1) = 0$ true.

$x = 180 n, 180 + 180 n, 210 + 360 n, 330 + 360 n$ , for any integer

$n$

Step 7

Consolidate

$180 n$ and

$180 + 180 n$ to

$180 n$ .

$x = 180 n, 210 + 360 n, 330 + 360 n$ , for any integer

$n$