Solve for x tan(x+pi/6)=-3

$\tan (x + \frac{π}{6}) = - 3$

Step 1

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

$x + \frac{π}{6} = \arctan (- 3)$

Step 2

Simplify the right side.

Evaluate $\arctan (- 3)$ .

$x + \frac{π}{6} = - 1.24904577$

Step 3

Move all terms not containing $x$ to the right side of the equation.

Subtract $\frac{π}{6}$ from both sides of the equation.

$x = - 1.24904577 - \frac{π}{6}$

Subtract $\frac{π}{6}$ from $- 1.24904577$ .

$x = - 1.77264454$

Step 4

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.

$x + \frac{π}{6} = - 1.24904577 - (3.14159265)$

Step 5

Simplify the expression to find the second solution.

Add $2 π$ to $- 1.24904577 - (3.14159265)$ .

$x + \frac{π}{6} = - 1.24904577 - (3.14159265) + 2 π$

The resulting angle of $1.89254688$ is positive and coterminal with $- 1.24904577 - (3.14159265)$ .

$x + \frac{π}{6} = 1.89254688$

Move all terms not containing $x$ to the right side of the equation.

Subtract $\frac{π}{6}$ from both sides of the equation.

$x = 1.89254688 - \frac{π}{6}$

Subtract $\frac{π}{6}$ from $1.89254688$ .

$x = 1.3689481$

Step 6

Find the period of $\tan (x + \frac{π}{6})$ .

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

$\frac{π}{| b |}$

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

$\frac{π}{| 1 |}$

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

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

Step 7

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

Add $π$ to $- 1.77264454$ to find the positive angle.

$- 1.77264454 + π$

Replace with decimal approximation.

$3.14159265 - 1.77264454$

Subtract $1.77264454$ from $3.14159265$ .

$1.3689481$

List the new angles.

$x = 1.3689481$

Step 8

The period of the $\tan (x + \frac{π}{6})$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = 1.3689481 + π n, 1.3689481 + π n$ , for any integer $n$

Step 9

Consolidate $1.3689481 + π n$ and $1.3689481 + π n$ to $1.3689481 + π n$ .

$x = 1.3689481 + π n$ , for any integer $n$

Solve for x tan(x+pi/6)=-3

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