Answer:
A + B must lie between 90 and 450 degrees.
Step-by-step explanation:
Since sin A and cos B are both negative, A and B must lie in the third quadrant or the fourth quadrant. In the third quadrant, sin A and cos B are both negative, while in the fourth quadrant, sin A is negative and cos B is positive.
Let's consider the case where A and B are both in the third quadrant. In this case, both A and B are between 180 and 270 degrees. Since A + B is the sum of two angles in this range, it follows that A + B must be between 360 and 450 degrees.
Now let's consider the case where A is in the fourth quadrant and B is in the third quadrant. In this case, A is between 270 and 360 degrees, while B is between 180 and 270 degrees. Since sin A is negative and cos B is positive, it follows that cos A and sin B are both negative. Using the identities sin(180 - x) = sin x and cos(180 - x) = -cos x, we can write:
sin A + cos B = sin(180 - (180 - A) - B) + cos(180 - (180 - A) - B)
= sin(180 - A) - cos(180 - B)
= -sin A - cos B
Therefore, A + B is the difference of two angles in the third quadrant, and it must be between 90 and 180 degrees.
Putting the two cases together, we see that A + B must lie between 90 and 450 degrees.