Trigonometry Examples

$5 \sin (x) = 2 \cos (x)$

Step 1

Divide each term in the equation by $\cos (x)$ .

$\frac{5 \sin (x)}{\cos (x)} = \frac{2 \cos (x)}{\cos (x)}$

Step 2

Separate fractions.

$\frac{5}{1} \cdot \frac{\sin (x)}{\cos (x)} = \frac{2 \cos (x)}{\cos (x)}$

Step 3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{5}{1} \cdot \tan (x) = \frac{2 \cos (x)}{\cos (x)}$

Step 4

Divide $5$ by $1$ .

$5 \tan (x) = \frac{2 \cos (x)}{\cos (x)}$

Step 5

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

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Step 5.1

Cancel the common factor.

$5 \tan (x) = \frac{2 \cos (x)}{\cos (x)}$

Step 5.2

Divide $2$ by $1$ .

$5 \tan (x) = 2$

Step 6

Divide each term in $5 \tan (x) = 2$ by $5$ and simplify.

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Step 6.1

Divide each term in $5 \tan (x) = 2$ by $5$ .

$\frac{5 \tan (x)}{5} = \frac{2}{5}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $5$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{5 \tan (x)}{5} = \frac{2}{5}$

Step 6.2.1.2

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

$\tan (x) = \frac{2}{5}$

Step 7

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

$x = \arctan (\frac{2}{5})$

Step 8

Simplify the right side.

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Step 8.1

Evaluate $\arctan (\frac{2}{5})$ .

$x = 0.38050637$

Step 9

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = (3.14159265) + 0.38050637$

Step 10

Solve for $x$ .

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Step 10.1

Remove parentheses.

$x = 3.14159265 + 0.38050637$

Step 10.2

Remove parentheses.

$x = (3.14159265) + 0.38050637$

Step 10.3

Add $3.14159265$ and $0.38050637$ .

$x = 3.52209903$

Step 11

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

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Step 11.1

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

$\frac{π}{| b |}$

Step 11.2

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

$\frac{π}{| 1 |}$

Step 11.3

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

$\frac{π}{1}$

Step 11.4

Divide $π$ by $1$ .

$π$

Step 12

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

$x = 0.38050637 + π n, 3.52209903 + π n$ , for any integer $n$

Step 13

Consolidate $0.38050637 + π n$ and $3.52209903 + π n$ to $0.38050637 + π n$ .

$x = 0.38050637 + π n$ , for any integer $n$