Use the information given below to find cos(alpha - beta)

Question

sin alpha 12/13 , with alpha in quadrant 1

tan beta 4/3, with beta in quadrant 3

*exact answer*

Answer 1

So we have sin α = 12/13, with α in quadrant 1.

Fortunately 12 and 13 are a leg and hypotenuse of a well-known Pythagorean triangle (5-12-13), so it follows that the cosine must be 5/13.

Similarly, tan β = 4/3. Here again we have two sides of a Pythagorean triangle, 3-4-5 this time. But both the sine and the cosine are negative in quadrant 3, so we have sin β = -4/5 and cos β = -3/5.

Then all we need is the formula for cos (α - β) = cos α cos β + sin α sin β.