Solve for A in Degrees -2tan(A)+2=4tan(A)+6

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Trigonometry

$- 2 \tan (A) + 2 = 4 \tan (A) + 6$

Move all terms containing $\tan (A)$ to the left side of the equation.

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Subtract $4 \tan (A)$ from both sides of the equation.

$- 2 \tan (A) + 2 - 4 \tan (A) = 6$

Subtract $4 \tan (A)$ from $- 2 \tan (A)$ .

$- 6 \tan (A) + 2 = 6$

Move all terms not containing $\tan (A)$ to the right side of the equation.

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Subtract $2$ from both sides of the equation.

$- 6 \tan (A) = 6 - 2$

Subtract $2$ from $6$ .

$- 6 \tan (A) = 4$

Divide each term in $- 6 \tan (A) = 4$ by $- 6$ and simplify.

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Divide each term in $- 6 \tan (A) = 4$ by $- 6$ .

$\frac{- 6 \tan (A)}{- 6} = \frac{4}{- 6}$

Simplify the left side.

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Cancel the common factor of $- 6$ .

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Cancel the common factor.

$\frac{- 6 \tan (A)}{- 6} = \frac{4}{- 6}$

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

$\tan (A) = \frac{4}{- 6}$

Simplify the right side.

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Cancel the common factor of $4$ and $- 6$ .

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Factor $2$ out of $4$ .

$\tan (A) = \frac{2 (2)}{- 6}$

Cancel the common factors.

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Factor $2$ out of $- 6$ .

$\tan (A) = \frac{2 \cdot 2}{2 \cdot - 3}$

Cancel the common factor.

$\tan (A) = \frac{2 \cdot 2}{2 \cdot - 3}$

Rewrite the expression.

$\tan (A) = \frac{2}{- 3}$

Move the negative in front of the fraction.

$\tan (A) = - \frac{2}{3}$

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

$A = \arctan (- \frac{2}{3})$

Simplify the right side.

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

$A = - 33.69006752$

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.

$A = - 33.69006752 - 180$

Simplify the expression to find the second solution.

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Add $360 °$ to $- 33.69006752 - 180 °$ .

$A = - 33.69006752 - 180 ° + 360 °$

The resulting angle of $146.30993247 °$ is positive and coterminal with $- 33.69006752 - 180$ .

$A = 146.30993247 °$

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

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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 $- 33.69006752$ to find the positive angle.

$- 33.69006752 + 180$

Subtract $33.69006752$ from $180$ .

$146.30993247$

List the new angles.

$A = 146.30993247$

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

$A = 146.30993247 + 180 n, 146.30993247 + 180 n$ , for any integer $n$

Consolidate $146.30993247 + 180 n$ and $146.30993247 + 180 n$ to $146.30993247 + 180 n$ .

$A = 146.30993247 + 180 n$ , for any integer $n$

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