Solve for B in Degrees 5tan(B)+7=2tan(B)+4

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Trigonometry

$5 \tan (B) + 7 = 2 \tan (B) + 4$

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

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Subtract $2 \tan (B)$ from both sides of the equation.

$5 \tan (B) + 7 - 2 \tan (B) = 4$

Subtract $2 \tan (B)$ from $5 \tan (B)$ .

$3 \tan (B) + 7 = 4$

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

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Subtract $7$ from both sides of the equation.

$3 \tan (B) = 4 - 7$

Subtract $7$ from $4$ .

$3 \tan (B) = - 3$

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

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

$\frac{3 \tan (B)}{3} = \frac{- 3}{3}$

Simplify the left side.

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Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 \tan (B)}{3} = \frac{- 3}{3}$

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

$\tan (B) = \frac{- 3}{3}$

Simplify the right side.

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Divide $- 3$ by $3$ .

$\tan (B) = - 1$

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

$B = \arctan (- 1)$

Simplify the right side.

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The exact value of $\arctan (- 1)$ is $- 45$ .

$B = - 45$

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.

$B = - 45 - 180$

Simplify the expression to find the second solution.

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Add $360 °$ to $- 45 - 180 °$ .

$B = - 45 - 180 ° + 360 °$

The resulting angle of $135 °$ is positive and coterminal with $- 45 - 180$ .

$B = 135 °$

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

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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 $- 45$ to find the positive angle.

$- 45 + 180$

Subtract $45$ from $180$ .

$135$

List the new angles.

$B = 135$

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

$B = 135 + 180 n, 135 + 180 n$ , for any integer $n$

Consolidate the answers.

$B = 135 + 180 n$ , for any integer $n$

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