Trigonometry Examples

$3 \cos (x) + 4 \sin (x) = 0$

Step 1

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

$\frac{3 \cos (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2

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

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Step 2.1

Cancel the common factor.

$\frac{3 \cos (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2.2

Divide $3$ by $1$ .

$3 + \frac{4 \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 3

Separate fractions.

$3 + \frac{4}{1} \cdot \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 4

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

$3 + \frac{4}{1} \cdot \tan (x) = \frac{0}{\cos (x)}$

Step 5

Divide $4$ by $1$ .

$3 + 4 \tan (x) = \frac{0}{\cos (x)}$

Step 6

Separate fractions.

$3 + 4 \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 7

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

$3 + 4 \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 8

Divide $0$ by $1$ .

$3 + 4 \tan (x) = 0 \sec (x)$

Step 9

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

$3 + 4 \tan (x) = 0$

Step 10

Subtract $3$ from both sides of the equation.

$4 \tan (x) = - 3$

Step 11

Divide each term in $4 \tan (x) = - 3$ by $4$ and simplify.

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Step 11.1

Divide each term in $4 \tan (x) = - 3$ by $4$ .

$\frac{4 \tan (x)}{4} = \frac{- 3}{4}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $4$ .

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Step 11.2.1.1

Cancel the common factor.

$\frac{4 \tan (x)}{4} = \frac{- 3}{4}$

Step 11.2.1.2

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

$\tan (x) = \frac{- 3}{4}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Move the negative in front of the fraction.

$\tan (x) = - \frac{3}{4}$

Step 12

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

$x = \arctan (- \frac{3}{4})$

Step 13

Simplify the right side.

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Step 13.1

Evaluate $\arctan (- \frac{3}{4})$ .

$x = - 0.6435011$

Step 14

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

$x = - 0.6435011 - (3.14159265)$

Step 15

Simplify the expression to find the second solution.

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Step 15.1

Add $2 π$ to $- 0.6435011 - (3.14159265)$ .

$x = - 0.6435011 - (3.14159265) + 2 π$

Step 15.2

The resulting angle of $2.49809154$ is positive and coterminal with $- 0.6435011 - (3.14159265)$ .

$x = 2.49809154$

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

Add $π$ to every negative angle to get positive angles.

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Step 17.1

Add $π$ to $- 0.6435011$ to find the positive angle.

$- 0.6435011 + π$

Step 17.2

Replace with decimal approximation.

$3.14159265 - 0.6435011$

Step 17.3

Subtract $0.6435011$ from $3.14159265$ .

$2.49809154$

Step 17.4

List the new angles.

$x = 2.49809154$

Step 18

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

$x = 2.49809154 + π n, 2.49809154 + π n$ , for any integer $n$

Step 19

Consolidate $2.49809154 + π n$ and $2.49809154 + π n$ to $2.49809154 + π n$ .

$x = 2.49809154 + π n$ , for any integer $n$