Answer:
Correct answer: cos(α - β) = 140/221
Step-by-step explanation:
cosα = ±√(1 - sin²α) => cosα = ±√(1 - (-5/13)²) = ±√(1 - (25/169)
cosα = ±√(169 -25)/169 = ± √144/169
cosα = ± 12/13 since it is given that α (alpha) ends in the third quadrant
we choose cosα = - 12/13
we know that is tanβ = sinβ/cosβ = - 8/15 or sinβ : cosβ = 8 : 15 =>
sinβ = 8k and cosβ = 15k where k is the coefficient of proportionality
we know the basic trigonometric equality
sin²β + cos²β = 1 when we replace the coefficients we get
(8k)² + (15k)² = 1 => 64k² + 225k² = 1 => 289k² = 1 => k² = 1/289 =>
k = 1/17 now we get sinβ and cosβ
since it is given that β (beta) ends in the second quadrant
sinβ = 8/17 and cosβ = - 15/17
As we know it is:
cos(α - β) = cosα cosβ + sinα sinβ = - 12/13 · (- 15/17) + (- 5/13) · 8/17 =>
cos(α - β) = 180/221 - 40/221 = 140/221
cos(α - β) = 140/221
God is with you!!!
