If sin alpha = 5/13, sin beta = 4/5
0 ≤ alpha ≤ pi/2 and pi/2 ≤ beta ≤ pi
==> find
a) cos alpha
b) cos beta
c) sin (alpha-beta)
d) cos (alpha+beta)
e) sin(2alpha)
f) cos(2beta)
Any help would be much appreciated.
Thank you so much!
0 ≤ alpha ≤ pi/2 and pi/2 ≤ beta ≤ pi
==> find
a) cos alpha
b) cos beta
c) sin (alpha-beta)
d) cos (alpha+beta)
e) sin(2alpha)
f) cos(2beta)
Any help would be much appreciated.
Thank you so much!
-
cos α = 12/13
cos β = -3/5
sin(α - β) = sin α cos β - sin β cos α = (5/13)(-3/5) - (4/5)(12/13) = -63/65
cos(α + β) = cos α cos β - sin α sin β = (12/13)(-3/5) - (5/13)(4/5) = -56/65
sin(2α) = 2sin α cos α = 2(5/13)(12/13) = 120/169
cos(2β) = 2cos²β - 1 = 2(9/25) - 1 = -7/25
cos β = -3/5
sin(α - β) = sin α cos β - sin β cos α = (5/13)(-3/5) - (4/5)(12/13) = -63/65
cos(α + β) = cos α cos β - sin α sin β = (12/13)(-3/5) - (5/13)(4/5) = -56/65
sin(2α) = 2sin α cos α = 2(5/13)(12/13) = 120/169
cos(2β) = 2cos²β - 1 = 2(9/25) - 1 = -7/25
-
Well,
we have:
sin alpha = 5/13, sin beta = 4/5
0 ≤ alpha ≤ pi/2 -----------> alpha in Quadrant I -----> cos alpha = sqrt(1- 5^2/13^2) = 12/13
pi/2 ≤ beta ≤ pi ---> beta in Q II so cos beta< 0 and cos beta = - sqrt(1-4^2/5^2)= -3/5
finally :
cos alpha = 12/13 and
cos beta = - 3/5
and the rest is easy :
cos alpha = 12/13 , sin alpha=5/13 , cos beta = - 3/5 , sin beta = 4/5
sin (alpha-beta) = sin alpha cos beta - cos alpha sin beta = (-15 - 48)/65= - 63/65
cos (alpha+beta) = cos alpha cos beta - sin alpha sin beta= (-36 - 20)/65= -46/65
sin(2alpha) = 2 sin alpha cos alpha = 120/169
cos(2beta) = cos^2 beta - sin^2 beta = (9 - 16)/25 = - 7/25
hope it'll help !!
we have:
sin alpha = 5/13, sin beta = 4/5
0 ≤ alpha ≤ pi/2 -----------> alpha in Quadrant I -----> cos alpha = sqrt(1- 5^2/13^2) = 12/13
pi/2 ≤ beta ≤ pi ---> beta in Q II so cos beta< 0 and cos beta = - sqrt(1-4^2/5^2)= -3/5
finally :
cos alpha = 12/13 and
cos beta = - 3/5
and the rest is easy :
cos alpha = 12/13 , sin alpha=5/13 , cos beta = - 3/5 , sin beta = 4/5
sin (alpha-beta) = sin alpha cos beta - cos alpha sin beta = (-15 - 48)/65= - 63/65
cos (alpha+beta) = cos alpha cos beta - sin alpha sin beta= (-36 - 20)/65= -46/65
sin(2alpha) = 2 sin alpha cos alpha = 120/169
cos(2beta) = cos^2 beta - sin^2 beta = (9 - 16)/25 = - 7/25
hope it'll help !!