Find the value of sin(alpha-beta), cos(alpha-beta) and tan(alpha-beta), given cos alpha=-12/13, cot beta=24/7, alpha lies in quadrant one and beta in quadrant two.
Answers
Given : cos alpha=-12/13, cot beta=24/7 alpha lies in quadrant two and beta in quadrant three ( correction)
To Find : Sin(α - β) , Cos(α - β) , tan(α - β)
Solution:
alpha lies in quadrant two
cosα= -12/13
=> Sinα = + Ve as in 2nd Quadrant Sin is + Ve
Sin²α = 1 - Cos²α
=> Sinα = 5/13
=> tanα =Sinα /cosα = - 5/12
beta in quadrant three Sin - ve , Cos -ve
cotβ=24/7 => tanβ = 7/24
Sinβ = -7/25
cosβ = - 24/25
Sin(α - β) = SinαCosβ - CosαSinβ
= (5/13)(-24/25) - (-12/13)(-7/25)
= -120/325 - 84/325
= -204/325
Cos(α - β) = CosαCosβ + SinαSinβ
= (-12/13)(-24/25) + (5/13)(-7/25)
= 288/325 - 35/325
= 253/325
tan(α - β) = Sin(α - β) / Cos(α - β) = - 204/253
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