Trigonometry Examples

$3 \cos (x) \tan (x) = - 5 \tan (x)$

Step 1

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

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Step 1.1

Add parentheses.

$3 (\cos (x) \tan (x)) = - 5 \tan (x)$

Step 1.2

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

$3 (\tan (x) \cos (x)) = - 5 \tan (x)$

Step 1.3

Rewrite $3 \cos (x) \tan (x)$ in terms of sines and cosines.

$3 (\frac{\sin (x)}{\cos (x)} \cos (x)) = - 5 \tan (x)$

Step 1.4

Cancel the common factors.

$3 \sin (x) = - 5 \tan (x)$

Step 2

Divide each term in $3 \sin (x) = - 5 \tan (x)$ by $- 5 \tan (x)$ and simplify.

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Step 2.1

Divide each term in $3 \sin (x) = - 5 \tan (x)$ by $- 5 \tan (x)$ .

$\frac{3 \sin (x)}{- 5 \tan (x)} = \frac{- 5 \tan (x)}{- 5 \tan (x)}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Separate fractions.

$\frac{3}{- 5} \cdot \frac{\sin (x)}{\tan (x)} = \frac{- 5 \tan (x)}{- 5 \tan (x)}$

Step 2.2.2

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

$\frac{3}{- 5} \cdot \frac{\sin (x)}{\frac{\sin (x)}{\cos (x)}} = \frac{- 5 \tan (x)}{- 5 \tan (x)}$

Step 2.2.3

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

$\frac{3}{- 5} (\sin (x) \frac{\cos (x)}{\sin (x)}) = \frac{- 5 \tan (x)}{- 5 \tan (x)}$

Step 2.2.4

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

$\frac{3}{- 5} (\frac{\sin (x)}{1} \cdot \frac{\cos (x)}{\sin (x)}) = \frac{- 5 \tan (x)}{- 5 \tan (x)}$

Step 2.2.5

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

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Step 2.2.5.1

Cancel the common factor.

$\frac{3}{- 5} (\frac{\sin (x)}{1} \cdot \frac{\cos (x)}{\sin (x)}) = \frac{- 5 \tan (x)}{- 5 \tan (x)}$

Step 2.2.5.2

Rewrite the expression.

$\frac{3}{- 5} \cos (x) = \frac{- 5 \tan (x)}{- 5 \tan (x)}$

Step 2.2.6

Move the negative in front of the fraction.

$- \frac{3}{5} \cos (x) = \frac{- 5 \tan (x)}{- 5 \tan (x)}$

Step 2.2.7

Combine $\cos (x)$ and $\frac{3}{5}$ .

$- \frac{\cos (x) \cdot 3}{5} = \frac{- 5 \tan (x)}{- 5 \tan (x)}$

Step 2.2.8

Move $3$ to the left of $\cos (x)$ .

$- \frac{3 \cos (x)}{5} = \frac{- 5 \tan (x)}{- 5 \tan (x)}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $- 5$ .

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Step 2.3.1.1

Cancel the common factor.

$- \frac{3 \cos (x)}{5} = \frac{- 5 \tan (x)}{- 5 \tan (x)}$

Step 2.3.1.2

Rewrite the expression.

$- \frac{3 \cos (x)}{5} = \frac{\tan (x)}{\tan (x)}$

Step 2.3.2

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

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Step 2.3.2.1

Cancel the common factor.

$- \frac{3 \cos (x)}{5} = \frac{\tan (x)}{\tan (x)}$

Step 2.3.2.2

Rewrite the expression.

$- \frac{3 \cos (x)}{5} = 1$

Step 3

Multiply both sides of the equation by $- \frac{5}{3}$ .

$- \frac{5}{3} (- \frac{3 \cos (x)}{5}) = - \frac{5}{3} \cdot 1$

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 $- \frac{5}{3} (- \frac{3 \cos (x)}{5})$ .

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Step 4.1.1.1

Cancel the common factor of $5$ .

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Step 4.1.1.1.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$\frac{- 5}{3} (- \frac{3 \cos (x)}{5}) = - \frac{5}{3} \cdot 1$

Step 4.1.1.1.2

Move the leading negative in $- \frac{3 \cos (x)}{5}$ into the numerator.

$\frac{- 5}{3} \cdot \frac{- 3 \cos (x)}{5} = - \frac{5}{3} \cdot 1$

Step 4.1.1.1.3

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{3} \cdot \frac{- 3 \cos (x)}{5} = - \frac{5}{3} \cdot 1$

Step 4.1.1.1.4

Cancel the common factor.

$\frac{5 \cdot - 1}{3} \cdot \frac{- 3 \cos (x)}{5} = - \frac{5}{3} \cdot 1$

Step 4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 \cos (x)) = - \frac{5}{3} \cdot 1$

Step 4.1.1.2

Cancel the common factor of $3$ .

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Step 4.1.1.2.1

Factor $3$ out of $- 3 \cos (x)$ .

$\frac{- 1}{3} (3 (- \cos (x))) = - \frac{5}{3} \cdot 1$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- \cos (x))) = - \frac{5}{3} \cdot 1$

Step 4.1.1.2.3

Rewrite the expression.

$- - \cos (x) = - \frac{5}{3} \cdot 1$

Step 4.1.1.3

Multiply.

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Step 4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 \cos (x) = - \frac{5}{3} \cdot 1$

Step 4.1.1.3.2

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

$\cos (x) = - \frac{5}{3} \cdot 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 1$ by $1$ .

$\cos (x) = - \frac{5}{3}$

Step 5

The range of cosine is $- 1 \leq y \leq 1$ . Since $- \frac{5}{3}$ does not fall in this range, there is no solution.

No solution