Question

Angles α and β are angles in standard position such that:

α terminates in Quadrant III and sinα = -5/13
β terminates in Quadrant II and tanβ = -8/15

Find cos(α - β).

Answer 1

Correct answer: cos(α - β) = 140/221

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!!!