Trigonometry Examples

$7 \sin (x) = 5 \tan (x)$

Step 1

Divide each term in $7 \sin (x) = 5 \tan (x)$ by $5 \tan (x)$ and simplify.

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Step 1.1

Divide each term in $7 \sin (x) = 5 \tan (x)$ by $5 \tan (x)$ .

$\frac{7 \sin (x)}{5 \tan (x)} = \frac{5 \tan (x)}{5 \tan (x)}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Separate fractions.

$\frac{7}{5} \cdot \frac{\sin (x)}{\tan (x)} = \frac{5 \tan (x)}{5 \tan (x)}$

Step 1.2.2

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

$\frac{7}{5} \cdot \frac{\sin (x)}{\frac{\sin (x)}{\cos (x)}} = \frac{5 \tan (x)}{5 \tan (x)}$

Step 1.2.3

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

$\frac{7}{5} (\sin (x) \frac{\cos (x)}{\sin (x)}) = \frac{5 \tan (x)}{5 \tan (x)}$

Step 1.2.4

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

$\frac{7}{5} (\frac{\sin (x)}{1} \cdot \frac{\cos (x)}{\sin (x)}) = \frac{5 \tan (x)}{5 \tan (x)}$

Step 1.2.5

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

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Step 1.2.5.1

Cancel the common factor.

$\frac{7}{5} (\frac{\sin (x)}{1} \cdot \frac{\cos (x)}{\sin (x)}) = \frac{5 \tan (x)}{5 \tan (x)}$

Step 1.2.5.2

Rewrite the expression.

$\frac{7}{5} \cos (x) = \frac{5 \tan (x)}{5 \tan (x)}$

Step 1.2.6

Combine $\frac{7}{5}$ and $\cos (x)$ .

$\frac{7 \cos (x)}{5} = \frac{5 \tan (x)}{5 \tan (x)}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $5$ .

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Step 1.3.1.1

Cancel the common factor.

$\frac{7 \cos (x)}{5} = \frac{5 \tan (x)}{5 \tan (x)}$

Step 1.3.1.2

Rewrite the expression.

$\frac{7 \cos (x)}{5} = \frac{\tan (x)}{\tan (x)}$

Step 1.3.2

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

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Step 1.3.2.1

Cancel the common factor.

$\frac{7 \cos (x)}{5} = \frac{\tan (x)}{\tan (x)}$

Step 1.3.2.2

Rewrite the expression.

$\frac{7 \cos (x)}{5} = 1$

Step 2

Multiply both sides of the equation by $\frac{5}{7}$ .

$\frac{5}{7} \cdot \frac{7 \cos (x)}{5} = \frac{5}{7} \cdot 1$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{5}{7} \cdot \frac{7 \cos (x)}{5}$ .

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Step 3.1.1.1

Cancel the common factor of $5$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{5}{7} \cdot \frac{7 \cos (x)}{5} = \frac{5}{7} \cdot 1$

Step 3.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 \cos (x)) = \frac{5}{7} \cdot 1$

Step 3.1.1.2

Cancel the common factor of $7$ .

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Step 3.1.1.2.1

Factor $7$ out of $7 \cos (x)$ .

$\frac{1}{7} (7 \cos (x)) = \frac{5}{7} \cdot 1$

Step 3.1.1.2.2

Cancel the common factor.

$\frac{1}{7} (7 \cos (x)) = \frac{5}{7} \cdot 1$

Step 3.1.1.2.3

Rewrite the expression.

$\cos (x) = \frac{5}{7} \cdot 1$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $\frac{5}{7}$ by $1$ .

$\cos (x) = \frac{5}{7}$

Step 4

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

$x = \arccos (\frac{5}{7})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arccos (\frac{5}{7})$ .

$x = 0.77519337$

Step 6

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 (3.14159265) - 0.77519337$

Step 7

Simplify $2 (3.14159265) - 0.77519337$ .

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Step 7.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 0.77519337$

Step 7.2

Subtract $0.77519337$ from $6.2831853$ .

$x = 5.50799193$

Step 8

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

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Step 8.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 8.2

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

$\frac{2 π}{| 1 |}$

Step 8.3

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

$\frac{2 π}{1}$

Step 8.4

Divide $2 π$ by $1$ .

$2 π$

Step 9

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 0.77519337 + 2 π n, 5.50799193 + 2 π n$ , for any integer $n$