Find sin(a+b) if cos(a)=3/5, sin(b)= -4/5, a is in quadrant 1 and b is in quadrant 3.
cos^-1(cos(a)=3/5)cos^-1 => a = 53.13010235. sin-1(sin(b)= -4/5) => -53.13010235

a + b = 0, and sin(0) = 0. but my answer of 0 is marked wrong. what am i missing?

only for . The angle you found lies in quadrant 4, not 3. Subtract 90 degrees from the angle you found and you get an angle in quadrant 3 whose sine is still -4/5, so that

or equivalently about 216 degrees (which does lie in quadrant 3).

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Without looking for the exact values of and , we can make two observations:

Next, we find the values of and from the given values using the Pythagorean identity:

Then using the angle sum identity for sine, we get

to complete this square you have to rotate about the origin 270 degrees clockwise.