How do you evaluate the expression cos(u+v) given cosu=4/7 with 0<u<pi/2 and sinv=-9/10 with pi<v<(3pi)/2?

1 Answer
Jul 31, 2016

(9sqrt(33)-4sqrt(19))/70

Explanation:

Since cos(u+v)=cosucosv-sinusinv

let's calculate (1)sin u and (2)cos v:

(1) sinu=+-sqrt(1-cos^2u);

in the first quadrant (0< u < pi/2) sinu>0, then

sinu=+sqrt(1-(4/7)^2)=sqrt(1-16/49)=sqrt((49-16)/49)=sqrt(33)/7

(2) cos v=+-sqrt(1-sin^2v)

in the third quadrant (pi < v < (3pi)/2) cosv<0, then

cosv=-sqrt(1-(-9/10)^2)=-sqrt(1-81/100)=-sqrt((100-81)/100)=-sqrt(19)/10

Then

cos(u+v)=cosucosv-sinusinv=
=4/7(-sqrt(19)/10)-sqrt(33)/7(-9/10)=
=-(4sqrt(19))/70+(9sqrt(33))/70
=(9sqrt(33)-4sqrt(19))/70