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

Question

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

cosα = −5/13, with α in quadrant II

cosβ = 15/17, with β in quadrant IV

Give the exact answer, not a decimal approximation.

Answer 1

sin(∝+ß) = sin(∝) cos(ß) + cos(∝) sin(ß)

for cos(∝) = -5 / 13 and ∝ is in Q2, then sin(∝) = 12 / 13

for cos(ß) = 15 / 17 and ß is in Q4, then sin(ß) = -8 / 17

Therefore:

sin(∝+ß) = sin(∝) cos(ß) + cos(∝) sin(ß) = ( 12 / 13 ) ( 15 / 17 ) + ( -5 / 13 ) ( -8 / 17 )

= (180 + 40) / (13 * 17)

= 220 / 221

Answer 2

cos(α) = −5/13, with α in quadrant II equates to this:

with the red number being calculated from the Pythagorean Theorem

That means sin(α) = 12/13

cos(β) = 15/17, with β in quadrant IV equates to this:

again with the red number being calculated from the Pythagorean Theorem

That means sin(β) = -8/17

Using the angle addition equations, sin(α + β) = sin(α)cos(β) + cos(α)sin(β) so, plugging in the numbers we get sin(α + β) = (12/13)(15/17) + (-5/13)(-8/17) = 180/221 + 40/221 = 220/221