Trig Identities

Question

Given 𝑠𝑖𝑛 𝛼 = 7 and 𝑠𝑖𝑛 𝛽 = 3, 𝛼 and 𝛽 in quadrant II, find 25 5

(a) 𝑠𝑖𝑛(𝛼 + 𝛽), (b) 𝑡𝑎𝑛(𝛼 + 𝛽), and (c) the quadrant of 𝛼 + 𝛽.

Answer 1

Given sin α = 7/25 in the second quadrant, cos α = -24 / 25

Given sin β = 3/5 in the second quadrant, cos β = -4 / 5

sin (α + β) = sin α cos β + cos α sin β = -100 / 125 = -4 / 5

cos (α + β) = cos α cos β - sin α sin β = 75 / 125 = 3 / 5

tan (α + β) = sin (α + β) / cos (α + β) = -4 / 3

Since the cosine of α + β is positive and the sine negative, α + β must be in the fourth quadrant.