If α and β are two angles in Quadrant II such that tan α= -1/2 and tan β = -2/3, find cos(α+β)

Question

Work:
cos(α+β) = [ 1 - (tan α)(tan β) ] / [ 1 + (tan α)(tan β)]

cos(α+β) = [ 1 - (-1/2)(-2/3) ] / [ 1 + (-1/2)(-2/3)]

cos(α+β) = [ 1 - 1/3 ] / [ 1 + 1/3 ]

cos(α+β) = [ 2/3 ] / [ 4/3 ]

cos(α+β) = 1/2

is this right?

Answer 1

Unfortunately it doesn't look right, ad... from the very start.

Could you rechceck your source from where you got the expression?

[ 1 - (tan α)(tan β) ] / [ 1 + (tan α)(tan β)]
represents, according to me,
cos(α+β)/cos(α-β).

All is not lost, though.

From

1/cos²(x)
=sec²(x)
=1+tan²(x)

tan(A+B)=(tan(A)+tan(B))/(1-tan(A)tan(B))

You have ample ammunition to get the required answer.

Post your results for a check-up if you wish, or you can substitute arbitrary numbers for tan(α) and tan(β) to check by yourself.

Answer 2

ohh okay thanks!

Answer 3

tan^-1(1/2)=x=.463648
tan^-1(2/3)=y.588003
x+y+pi/2 = -z=2.62245
z=-2.62245
cos (z) = -.868243

and

tan a= -1/2
tan b= -2/3
tan (a+b)=(tan a +tan b)/(1-tan a.tan b)
so tan (a+b)= -7/4
sec^2 x-tan^2 x=1
so
sec (a+b)=√(1+49/16)
=√(65/16)
(a+b)=1/sec (a+b)
so cos(a+b) is positive
so cos(a+b)=.48˜.5

???

Answer 4

tan^-1(1/2)=x=.463648
α=tan_-1(-1/2)=π-x=2.67795

tan^-1(2/3)=y.588003
β=tan_-1(-2/3)=π-y=2.55359

α+β = 2.67795+2.55359 = 5.23154 rad. (serves as a check)
cos(α+β) = 0.49614 (as a check)

and

tan a= -1/2
tan b= -2/3
tan (a+b)=(tan a +tan b)/(1-tan a.tan b)
so tan (a+b)= -7/4
sec^2 x-tan^2 x=1
so
sec (a+b)=√(1+49/16)
=√(65/16)
Therefore
cos(α+β)
=√(16/65)
=4/√65
since α+β is in the 4th quadrant, cos(α+β)>0

Therefore
cos(α+β) = +4/√65

Answer 5

oh! thanks for your time!