Solve for θ in Degrees tan(theta)^2+6tan(theta)+8=0

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Trigonometry

$\tan^{2} (θ) + 6 \tan (θ) + 8 = 0$

Factor the left side of the equation.

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Let $u = \tan (θ)$ . Substitute $u$ for all occurrences of $\tan (θ)$ .

$u^{2} + 6 u + 8 = 0$

Factor $u^{2} + 6 u + 8$ using the AC method.

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Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $8$ and whose sum is $6$ .

$2, 4$

Write the factored form using these integers.

$(u + 2) (u + 4) = 0$

Replace all occurrences of $u$ with $\tan (θ)$ .

$(\tan (θ) + 2) (\tan (θ) + 4) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\tan (θ) + 2 = 0$

$\tan (θ) + 4 = 0$

Set $\tan (θ) + 2$ equal to $0$ and solve for $θ$ .

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Set $\tan (θ) + 2$ equal to $0$ .

$\tan (θ) + 2 = 0$

Solve $\tan (θ) + 2 = 0$ for $θ$ .

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Subtract $2$ from both sides of the equation.

$\tan (θ) = - 2$

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

$θ = \arctan (- 2)$

Simplify the right side.

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Evaluate $\arctan (- 2)$ .

$θ = - 63.43494882$

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the third quadrant.

$θ = - 63.43494882 - 180$

Simplify the expression to find the second solution.

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Add $360 °$ to $- 63.43494882 - 180 °$ .

$θ = - 63.43494882 - 180 ° + 360 °$

The resulting angle of $116.56505117 °$ is positive and coterminal with $- 63.43494882 - 180$ .

$θ = 116.56505117 °$

Find the period of $\tan (θ)$ .

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The period of the function can be calculated using $\frac{180}{| b |}$ .

$\frac{180}{| b |}$

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

$\frac{180}{| 1 |}$

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

$\frac{180}{1}$

Divide $180$ by $1$ .

$180$

Add $180$ to every negative angle to get positive angles.

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Add $180$ to $- 63.43494882$ to find the positive angle.

$- 63.43494882 + 180$

Subtract $63.43494882$ from $180$ .

$116.56505117$

List the new angles.

$θ = 116.56505117$

The period of the $\tan (θ)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$θ = 116.56505117 + 180 n, 116.56505117 + 180 n$ , for any integer $n$

Set $\tan (θ) + 4$ equal to $0$ and solve for $θ$ .

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Set $\tan (θ) + 4$ equal to $0$ .

$\tan (θ) + 4 = 0$

Solve $\tan (θ) + 4 = 0$ for $θ$ .

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Subtract $4$ from both sides of the equation.

$\tan (θ) = - 4$

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

$θ = \arctan (- 4)$

Simplify the right side.

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Evaluate $\arctan (- 4)$ .

$θ = - 75.96375653$

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the third quadrant.

$θ = - 75.96375653 - 180$

Simplify the expression to find the second solution.

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Add $360 °$ to $- 75.96375653 - 180 °$ .

$θ = - 75.96375653 - 180 ° + 360 °$

The resulting angle of $104.03624346 °$ is positive and coterminal with $- 75.96375653 - 180$ .

$θ = 104.03624346 °$

Find the period of $\tan (θ)$ .

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The period of the function can be calculated using $\frac{180}{| b |}$ .

$\frac{180}{| b |}$

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

$\frac{180}{| 1 |}$

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

$\frac{180}{1}$

Divide $180$ by $1$ .

$180$

Add $180$ to every negative angle to get positive angles.

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Add $180$ to $- 75.96375653$ to find the positive angle.

$- 75.96375653 + 180$

Subtract $75.96375653$ from $180$ .

$104.03624346$

List the new angles.

$θ = 104.03624346$

The period of the $\tan (θ)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$θ = 104.03624346 + 180 n, 104.03624346 + 180 n$ , for any integer $n$

The final solution is all the values that make $(\tan (θ) + 2) (\tan (θ) + 4) = 0$ true.

$θ = 116.56505117 + 180 n, 104.03624346 + 180 n, 104.03624346 + 180 n$ , for any integer $n$

Consolidate $104.03624346 + 180 n$ and $104.03624346 + 180 n$ to $104.03624346 + 180 n$ .

$θ = 116.56505117 + 180 n, 104.03624346 + 180 n$ , for any integer $n$

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