Trigonometry Examples

$6 \tan (x) + 5 = 0$

Step 1

Subtract $5$ from both sides of the equation.

$6 \tan (x) = - 5$

Step 2

Divide each term in $6 \tan (x) = - 5$ by $6$ and simplify.

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Step 2.1

Divide each term in $6 \tan (x) = - 5$ by $6$ .

$\frac{6 \tan (x)}{6} = \frac{- 5}{6}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $6$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{6 \tan (x)}{6} = \frac{- 5}{6}$

Step 2.2.1.2

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

$\tan (x) = \frac{- 5}{6}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$\tan (x) = - \frac{5}{6}$

Step 3

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

$x = \arctan (- \frac{5}{6})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arctan (- \frac{5}{6})$ .

$x = - 0.69473827$

Step 5

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.

$x = - 0.69473827 - (3.14159265)$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Add $2 π$ to $- 0.69473827 - (3.14159265)$ .

$x = - 0.69473827 - (3.14159265) + 2 π$

Step 6.2

The resulting angle of $2.44685437$ is positive and coterminal with $- 0.69473827 - (3.14159265)$ .

$x = 2.44685437$

Step 7

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

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Step 7.1

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

$\frac{π}{| b |}$

Step 7.2

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

$\frac{π}{| 1 |}$

Step 7.3

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

$\frac{π}{1}$

Step 7.4

Divide $π$ by $1$ .

$π$

Step 8

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

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Step 8.1

Add $π$ to $- 0.69473827$ to find the positive angle.

$- 0.69473827 + π$

Step 8.2

Replace with decimal approximation.

$3.14159265 - 0.69473827$

Step 8.3

Subtract $0.69473827$ from $3.14159265$ .

$2.44685437$

Step 8.4

List the new angles.

$x = 2.44685437$

Step 9

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

$x = 2.44685437 + π n, 2.44685437 + π n$ , for any integer $n$

Step 10

Consolidate $2.44685437 + π n$ and $2.44685437 + π n$ to $2.44685437 + π n$ .

$x = 2.44685437 + π n$ , for any integer $n$