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y=-3tan[(1)/(2)(x-pi)]-2

$y=-3 \tan \left[\frac{1}{2}(x-\pi)\right]-2$

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Full solution

Q.

y=-3 \tan \left[\frac{1}{2}(x-\pi)\right]-2

Isolate trigonometric function: Isolate the trigonometric function. $\newline$ $y + 2 = -3\tan\left[\frac{1}{2}(x - \pi)\right]$
Divide by $-3$ : Divide both sides by $-3$ to get the $\tan$ alone. $\newline$ $(y + 2) / -3 = \tan\left[(1/2)(x - \pi)\right]$
Solve for angle: Solve for the angle inside the tan function. $\newline$ Let $\theta = \frac{1}{2}(x - \pi)$ $\newline$ $\tan(\theta) = \frac{y + 2}{-3}$
Find inverse tan: Find the inverse tan of both sides to solve for $\theta$ . $\theta = \arctan\left(\frac{y + 2}{-3}\right)$
Substitute back for $\theta$ : Substitute back for $\theta$ to solve for $x$ . $\newline$ $(1/2)(x - \pi) = \arctan((y + 2) / -3)$
Multiply by $2$ : Multiply both sides by $2$ to solve for $x - \pi$ . $x - \pi = 2 \times \arctan\left(\frac{y + 2}{-3}\right)$
Add $\pi$ : Add $\pi$ to both sides to solve for $x$ . $\newline$ $x = \pi + 2 \cdot \arctan\left(\frac{y + 2}{-3}\right)$

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