1. The problem asks to solve the equation $$\tan x = -1.23$$ for $$x$$ in degrees, giving the general solution including the integer parameter $$k\in \mathbb{Z}$$. The CAST diagram helps determine the quadrants where tangent is negative. 2. Tangent is negative in the second and fourth quadrants. We first find the reference angle $$\alpha$$ such that $$\tan \alpha = 1.23$$. 3. Calculate the reference angle: $$\alpha = \arctan(1.23) \approx 50.19$$ degrees. 4. Using the CAST diagram: - In the second quadrant, $$x = 180 - \alpha = 180 - 50.19 = 129.81$$ degrees. - In the fourth quadrant, $$x = 360 - \alpha = 360 - 50.19 = 309.81$$ degrees. 5. The general solutions for $$\tan x = -1.23$$ are: $$x = 129.81 + 180k ; 309.81 + 180k, \quad k \in \mathbb{Z}$$ Final answer: $$x = 129.81 + 180k ; 309.81 + 180k, \quad k \in \mathbb{Z}$$