Cos a = -(4/5), a lies in quadrant 3, and sin(b) = (sqrt21)/5, B lies in quadrant 2. FIND: cos(a+b)

Question

I have an idea of how to solve this, I just want to make sure I'm correct

Answer 1

This is an application of the sum and difference formulas.

cos(A + B) = cos A cos B  − sin A sin B

Since they give you cos A, you have to find sin A. Draw a triangle in the 3rd quadrant. Label the adjacent side as -4 and the hypotenuse as 5 based on the cos A=-4/5. The opposite side must be -3, based on the Pythagorean Theorem. So sin A = -3/5.

Since they give you sin B, you have to solve for cos B. Draw a triangle in the 2nd quadrant. Label the opposite side as √21 and the hypotenuse as 5 based on the sin B=√21/5. Again using the Pythagorean Theorem, you can solve for the adjacent side must be -2, So cos B=-2/5

Now plugging all the values into the formula:

cos(A + B) = (-4/5) (-2/5) − (-3/5)(√21/5) Multiply the fractions

8/25+3√21/25 =

(8+3√21)/25