Given alpha = (12)/(-5) and 90^ alpha 180^.

Given $\tan \alpha = \frac{12}{-5}$ and $90^\circ \le \alpha \le 180^\circ$.
Since $\tan \alpha = \frac{y}{x}$, we have $y = 12$ and $x = -5$.
The angle $\alpha$ is in the second quadrant, which is consistent with $x$ being negative and $y$ being positive.
Now, calculate the hypotenuse $r$ using the Pythagorean theorem:

• Draw a Cartesian plane with the x-axis and y-axis.
• Mark the origin $(0,0)$.
• Plot the point $P(-5, 12)$ in the second quadrant.
• Draw a line segment from the origin to $P(-5, 12)$. This represents the hypotenuse $r=13$.
• Drop a perpendicular line from $P(-5, 12)$ to the x-axis, meeting it at $(-5, 0)$. This forms a right-angled triangle.
• Label the horizontal side as $x=-5$, the vertical side as $y=12$, and the hypotenuse as $r=13$.
• Indicate the angle $\alpha$ measured counter-clockwise from the positive x-axis to the line segment $OP$.
• Indicate the reference angle $\theta_R$ inside the triangle, between the line segment $OP$ and the negative x-axis.

Using the values from Step 1:

Now, calculate the sum:

First, find the reference angle $\theta_R$ using the absolute value of $\tan \alpha$:

Since the problem states "without the use of calculator", we express $\theta_R$ using $\arctan$:

As $\alpha$ is in the second quadrant ($90^\circ \le \alpha \le 180^\circ$), the relationship between $\alpha$ and $\theta_R$ is:

Substitute the expression for $\theta_R$: