Given that: sin α = 15/17, with α in QI, and sin(α + β) = 12/13, with α + β in QII, find sin β and cos β?
Given that:
sin α = 15/17, with α in QI, and sin(α + β) = 12/13, with α + β in QII, find sin β and cos β
Given that:
sin α = 15/17, with α in QI, and sin(α + β) = 12/13, with α + β in QII, find sin β and cos β
1 Answer
# cos beta = 140/221 \ \ # and# \ \ sin beta= 171/221 #
Explanation:
Using
# cos^2 alpha =1 - sin^2 alpha #
# \ \ \ \ \ \ \ \ \ = 1-(15/17)^2 #
# \ \ \ \ \ \ \ \ \ = 1-225/289 #
# \ \ \ \ \ \ \ \ \ = 64/289 #
# :. cos alpha = sqrt(64/289) = +- 8/17 #
Knowing that
# cos alpha=8/17 # ..... [A]
Similarly, we know that:
# cos^2 (alpha+beta) =1 - sin^2 (alpha+beta) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =1 - (12/13)^2 #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =1 - 144/169 #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =25/169 #
# :. cos (alpha+beta) = sqrt(25/169) = +- 5/13 #
Knowing that
# cos(alpha+beta) = -5/13 # ..... [B]
Now, we can use the sum of angle formula:
# sin(A+B) = sinAcosB+cosAsinB #
# cos(A+B) = cosAcosB-sinAsinB #
From which we get, using [A] and [B], that:
# sin(alpha+beta)= sin alpha cos beta+cos alpha sin beta #
# :. 12/13= 15/17 cos beta + 8/17 sin beta # ..... [C]
And:
# cos(alpha+beta)= cos alpha cos beta - sin alpha sin beta #
# :. -5/13 = 8/17 cos beta - 15/17 sin beta # ..... [D]
We now have two simultaneous equations [C] and [D] in
# cos beta = 140/221 \ \ # and# \ \ sin beta= 171/221 #