If sin A=2/3 cos B=3/4, angle A is in quadrant 2 and angle B is in quadrant 4. How do you evaluate sin(A-B) without find A and B?
Please see the explanation.
We will need the value of
We are given that A is in the 2nd quadrant, therefore, we chose the negative value for the cosine:
We will need the
We are given that B is in the 4th quadrant, therefore, we chose the negative value for the sine:
We have all of the values that we need to use the identity:
The easiest way I can think of is to use the Pythagorean Identity to first find both
The trigonometric Pythagorean identity tells us that
which follows directly from the geometric identity for right triangles and the fact that
Anyway, if we know
Because we know
Similar reasoning can be done with
Now that we have all the necessary elements, we can use the sum/difference angle identity:
Hope this helps!