Trigonometry Examples

$5 \cos (x) \tan (x) = 9 \tan (x)$

Step 1

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

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Step 1.1

Add parentheses.

$5 (\cos (x) \tan (x)) = 9 \tan (x)$

Step 1.2

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

$5 (\tan (x) \cos (x)) = 9 \tan (x)$

Step 1.3

Rewrite $5 \cos (x) \tan (x)$ in terms of sines and cosines.

$5 (\frac{\sin (x)}{\cos (x)} \cos (x)) = 9 \tan (x)$

Step 1.4

Cancel the common factors.

$5 \sin (x) = 9 \tan (x)$

Step 2

Divide each term in $5 \sin (x) = 9 \tan (x)$ by $9 \tan (x)$ and simplify.

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Step 2.1

Divide each term in $5 \sin (x) = 9 \tan (x)$ by $9 \tan (x)$ .

$\frac{5 \sin (x)}{9 \tan (x)} = \frac{9 \tan (x)}{9 \tan (x)}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Separate fractions.

$\frac{5}{9} \cdot \frac{\sin (x)}{\tan (x)} = \frac{9 \tan (x)}{9 \tan (x)}$

Step 2.2.2

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

$\frac{5}{9} \cdot \frac{\sin (x)}{\frac{\sin (x)}{\cos (x)}} = \frac{9 \tan (x)}{9 \tan (x)}$

Step 2.2.3

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

$\frac{5}{9} (\sin (x) \frac{\cos (x)}{\sin (x)}) = \frac{9 \tan (x)}{9 \tan (x)}$

Step 2.2.4

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

$\frac{5}{9} (\frac{\sin (x)}{1} \cdot \frac{\cos (x)}{\sin (x)}) = \frac{9 \tan (x)}{9 \tan (x)}$

Step 2.2.5

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

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Step 2.2.5.1

Cancel the common factor.

$\frac{5}{9} (\frac{\sin (x)}{1} \cdot \frac{\cos (x)}{\sin (x)}) = \frac{9 \tan (x)}{9 \tan (x)}$

Step 2.2.5.2

Rewrite the expression.

$\frac{5}{9} \cos (x) = \frac{9 \tan (x)}{9 \tan (x)}$

Step 2.2.6

Combine $\frac{5}{9}$ and $\cos (x)$ .

$\frac{5 \cos (x)}{9} = \frac{9 \tan (x)}{9 \tan (x)}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $9$ .

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Step 2.3.1.1

Cancel the common factor.

$\frac{5 \cos (x)}{9} = \frac{9 \tan (x)}{9 \tan (x)}$

Step 2.3.1.2

Rewrite the expression.

$\frac{5 \cos (x)}{9} = \frac{\tan (x)}{\tan (x)}$

Step 2.3.2

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

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Step 2.3.2.1

Cancel the common factor.

$\frac{5 \cos (x)}{9} = \frac{\tan (x)}{\tan (x)}$

Step 2.3.2.2

Rewrite the expression.

$\frac{5 \cos (x)}{9} = 1$

Step 3

Multiply both sides of the equation by $\frac{9}{5}$ .

$\frac{9}{5} \cdot \frac{5 \cos (x)}{9} = \frac{9}{5} \cdot 1$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{9}{5} \cdot \frac{5 \cos (x)}{9}$ .

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Step 4.1.1.1

Cancel the common factor of $9$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{9}{5} \cdot \frac{5 \cos (x)}{9} = \frac{9}{5} \cdot 1$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 \cos (x)) = \frac{9}{5} \cdot 1$

Step 4.1.1.2

Cancel the common factor of $5$ .

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Step 4.1.1.2.1

Factor $5$ out of $5 \cos (x)$ .

$\frac{1}{5} (5 \cos (x)) = \frac{9}{5} \cdot 1$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 \cos (x)) = \frac{9}{5} \cdot 1$

Step 4.1.1.2.3

Rewrite the expression.

$\cos (x) = \frac{9}{5} \cdot 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $\frac{9}{5}$ by $1$ .

$\cos (x) = \frac{9}{5}$

Step 5

The range of cosine is $- 1 \leq y \leq 1$ . Since $\frac{9}{5}$ does not fall in this range, there is no solution.

No solution