Trigonometry Examples

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

Step 1

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

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Step 1.1

Add parentheses.

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

Step 1.2

Reorder $\cos (x)$ and $\tan (x)$ .

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

Step 1.3

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

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

Step 1.4

Cancel the common factors.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Separate fractions.

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

Step 2.2.2

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

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

Step 2.2.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 2.2.4

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 2.2.5

Cancel the common factor of $\sin (x)$ .

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Step 2.2.5.1

Cancel the common factor.

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

Step 2.2.5.2

Rewrite the expression.

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

Step 2.2.6

Divide $8$ by $- 1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Dividing two negative values results in a positive value.

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

Step 2.3.2

Cancel the common factor of $\tan (x)$ .

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Step 2.3.2.1

Cancel the common factor.

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

Step 2.3.2.2

Rewrite the expression.

$- 8 \cos (x) = 1$

Step 3

Divide each term in $- 8 \cos (x) = 1$ by $- 8$ and simplify.

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Step 3.1

Divide each term in $- 8 \cos (x) = 1$ by $- 8$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 8$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

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

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Simplify the right side.

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Step 5.1

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

$x = 97.18075578$

Step 6

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $360$ to find the solution in the third quadrant.

$x = 360 - 97.18075578$

Step 7

Subtract $97.18075578$ from $360$ .

$x = 262.81924421$

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

$\frac{360}{1}$

Step 8.4

Divide $360$ by $1$ .

$360$

Step 9

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

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