Precalculus Examples

$3 \sin (x) - 2 \cos (x) = 0$

Step 1

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

$\frac{3 \sin (x)}{\cos (x)} + \frac{- 2 \cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2

Separate fractions.

$\frac{3}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{- 2 \cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 3

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

$\frac{3}{1} \cdot \tan (x) + \frac{- 2 \cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 4

Divide $3$ by $1$ .

$3 \tan (x) + \frac{- 2 \cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 5

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

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Step 5.1

Cancel the common factor.

$3 \tan (x) + \frac{- 2 \cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 5.2

Divide $- 2$ by $1$ .

$3 \tan (x) - 2 = \frac{0}{\cos (x)}$

Step 6

Separate fractions.

$3 \tan (x) - 2 = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$3 \tan (x) - 2 = \frac{0}{1} \cdot \sec (x)$

Step 8

Divide $0$ by $1$ .

$3 \tan (x) - 2 = 0 \sec (x)$

Step 9

Multiply $0$ by $\sec (x)$ .

$3 \tan (x) - 2 = 0$

Step 10

Add $2$ to both sides of the equation.

$3 \tan (x) = 2$

Step 11

Divide each term in $3 \tan (x) = 2$ by $3$ and simplify.

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Step 11.1

Divide each term in $3 \tan (x) = 2$ by $3$ .

$\frac{3 \tan (x)}{3} = \frac{2}{3}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $3$ .

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Step 11.2.1.1

Cancel the common factor.

$\frac{3 \tan (x)}{3} = \frac{2}{3}$

Step 11.2.1.2

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

$\tan (x) = \frac{2}{3}$

Step 12

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

$x = \arctan (\frac{2}{3})$

Step 13

Simplify the right side.

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Step 13.1

Evaluate $\arctan (\frac{2}{3})$ .

$x = 0.5880026$

Step 14

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.

$x = (3.14159265) + 0.5880026$

Step 15

Solve for $x$ .

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Step 15.1

Remove parentheses.

$x = 3.14159265 + 0.5880026$

Step 15.2

Remove parentheses.

$x = (3.14159265) + 0.5880026$

Step 15.3

Add $3.14159265$ and $0.5880026$ .

$x = 3.72959525$

Step 16

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

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Step 16.1

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

$\frac{π}{| b |}$

Step 16.2

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

$\frac{π}{| 1 |}$

Step 16.3

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

$\frac{π}{1}$

Step 16.4

Divide $π$ by $1$ .

$π$

Step 17

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

$x = 0.5880026 + π n, 3.72959525 + π n$ , for any integer $n$

Step 18

Consolidate $0.5880026 + π n$ and $3.72959525 + π n$ to $0.5880026 + π n$ .

$x = 0.5880026 + π n$ , for any integer $n$