Use the information given below to find sin(α+β

Question

cos α = -5/13, with α in quadrant 3

cos β = 15/17, with β in quadrant 4

Give the exact answer, not a decimal approximation.

Answer 1

To find sin(α + β) we need to use the angle addition identity. It goes like this:

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

Obviously, to use this, we need 4 things (2 of which are given to us):

sin(α):

cos(β): 15/17

cos(α): -5/13

sin(β)

To find the other two things, we need to draw the triangles that were given to us. Like this:

To calculate the missing sides, use the Pythagorean Theorem.

For the triangle with angle α, the 3rd side is √(13² - 5²) = 12. But the sign of the 12 MUST be negative because it's in Q3. That means sin(α) = -12/13

For the triangle with angle β, the 3rd side is √(17² - 15²) = 8. But the sign of the 8 MUST be negative because it's in Q4. That means sin(β) = -8/17

So now, just plug in the numbers:

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

sin(α + β) = (-12/13)(15/17) + (-5/13)(-8/17)

multiply out and simplify