Trigonometry Examples

8 \cos (x) \tan (x) = - \tan (x)

$8 \cos (x) \tan (x) = - \tan (x)$

Step 1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.1

Add parentheses.

8 (\cos (x) \tan (x)) = - \tan (x)

$8 (\cos (x) \tan (x)) = - \tan (x)$

Step 1.2

Reorder

\cos (x)

$\cos (x)$ and

\tan (x)

$\tan (x)$ .

8 (\tan (x) \cos (x)) = - \tan (x)

$8 (\tan (x) \cos (x)) = - \tan (x)$

Step 1.3

Rewrite

8 \cos (x) \tan (x)

$8 \cos (x) \tan (x)$ in terms of sines and cosines.

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

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

Step 1.4

Cancel the common factors.

8 \sin (x) = - \tan (x)

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

8 \sin (x) = - \tan (x)

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

Step 2

Divide each term in

8 \sin (x) = - \tan (x)

$8 \sin (x) = - \tan (x)$ by

- \tan (x)

$- \tan (x)$ and simplify.

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Step 2.1

Divide each term in

8 \sin (x) = - \tan (x)

$8 \sin (x) = - \tan (x)$ by

- \tan (x)

$- \tan (x)$ .

\frac{8 \sin (x)}{- \tan (x)} = \frac{- \tan (x)}{- \tan (x)}

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Separate fractions.

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

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

Step 2.2.2

Rewrite

\tan (x)

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

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

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

Step 2.2.3

Multiply by the reciprocal of the fraction to divide by

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

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

\frac{8}{- 1} (\sin (x) \frac{\cos (x)}{\sin (x)}) = \frac{- \tan (x)}{- \tan (x)}

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

Step 2.2.4

Write

\sin (x)

$\sin (x)$ as a fraction with denominator

1

$1$ .

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

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

Step 2.2.5

Cancel the common factor of

\sin (x)

$\sin (x)$ .

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Step 2.2.5.1

Cancel the common factor.

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

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

Step 2.2.5.2

Rewrite the expression.

\frac{8}{- 1} \cos (x) = \frac{- \tan (x)}{- \tan (x)}

$\frac{8}{- 1} \cos (x) = \frac{- \tan (x)}{- \tan (x)}$

\frac{8}{- 1} \cos (x) = \frac{- \tan (x)}{- \tan (x)}

$\frac{8}{- 1} \cos (x) = \frac{- \tan (x)}{- \tan (x)}$

Step 2.2.6

Divide

8

$8$ by

- 1

$- 1$ .

- 8 \cos (x) = \frac{- \tan (x)}{- \tan (x)}

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

- 8 \cos (x) = \frac{- \tan (x)}{- \tan (x)}

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Dividing two negative values results in a positive value.

- 8 \cos (x) = \frac{\tan (x)}{\tan (x)}

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

Step 2.3.2

Cancel the common factor of

\tan (x)

$\tan (x)$ .

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Step 2.3.2.1

Cancel the common factor.

- 8 \cos (x) = \frac{\tan (x)}{\tan (x)}

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

Step 2.3.2.2

Rewrite the expression.

- 8 \cos (x) = 1

$- 8 \cos (x) = 1$

- 8 \cos (x) = 1

$- 8 \cos (x) = 1$

- 8 \cos (x) = 1

$- 8 \cos (x) = 1$

- 8 \cos (x) = 1

$- 8 \cos (x) = 1$

Step 3

Divide each term in

- 8 \cos (x) = 1

$- 8 \cos (x) = 1$ by

- 8

$- 8$ and simplify.

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Step 3.1

Divide each term in

- 8 \cos (x) = 1

$- 8 \cos (x) = 1$ by

- 8

$- 8$ .

\frac{- 8 \cos (x)}{- 8} = \frac{1}{- 8}

$\frac{- 8 \cos (x)}{- 8} = \frac{1}{- 8}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

- 8

$- 8$ .

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Step 3.2.1.1

Cancel the common factor.

\frac{- 8 \cos (x)}{- 8} = \frac{1}{- 8}

$\frac{- 8 \cos (x)}{- 8} = \frac{1}{- 8}$

Step 3.2.1.2

Divide

\cos (x)

$\cos (x)$ by

1

$1$ .

\cos (x) = \frac{1}{- 8}

$\cos (x) = \frac{1}{- 8}$

\cos (x) = \frac{1}{- 8}

$\cos (x) = \frac{1}{- 8}$

\cos (x) = \frac{1}{- 8}

$\cos (x) = \frac{1}{- 8}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

\cos (x) = - \frac{1}{8}

$\cos (x) = - \frac{1}{8}$

\cos (x) = - \frac{1}{8}

$\cos (x) = - \frac{1}{8}$

\cos (x) = - \frac{1}{8}

$\cos (x) = - \frac{1}{8}$

Step 4

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

x

$x$ from inside the cosine.

x = \arccos (- \frac{1}{8})

$x = \arccos (- \frac{1}{8})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate

\arccos (- \frac{1}{8})

$\arccos (- \frac{1}{8})$ .

x = 97.18075578

$x = 97.18075578$

x = 97.18075578

$x = 97.18075578$

Step 6

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from

360

$360$ to find the solution in the third quadrant.

x = 360 - 97.18075578

$x = 360 - 97.18075578$

Step 7

Subtract

97.18075578

$97.18075578$ from

360

$360$ .

x = 262.81924421

$x = 262.81924421$

Step 8

Find the period of

\cos (x)

$\cos (x)$ .

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Step 8.1

The period of the function can be calculated using

\frac{360}{| b |}

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

\frac{360}{| b |}

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

Step 8.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{360}{| 1 |}

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

Step 8.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{360}{1}

$\frac{360}{1}$

Step 8.4

Divide

360

$360$ by

1

$1$ .

360

$360$

360

$360$

Step 9

The period of the

\cos (x)

$\cos (x)$ function is

360

$360$ so values will repeat every

360

$360$ degrees in both directions.

x = 97.18075578 + 360 n, 262.81924421 + 360 n

$x = 97.18075578 + 360 n, 262.81924421 + 360 n$ , for any integer

n

$n$