How do you evaluate the expression cos(u-v) given sinu=3/5 with pi/2<u<p and cosv=-5/6 with pi<v<(3pi)/2?

1 Answer
Sep 27, 2017

see below

Explanation:

Use the composite argument formula cos (u-v)=cosucosv-sinusinv.

But first we need to find cos u and sin v. Since u is in Quadrant II and sin u= 3/5=(opposite)/(hypote n use we can find the adjacent side by using pythagorean theorem.

That is,
a^2=c^2-b^2

a=sqrt(5^2-3^2)=sqrt(25-9)=sqrt16=4 but since u is in quadrant two then a=-4. Hence, cos u=-4/5.

Likewise, we need to find sin v and v is in Quadrant III. So, to find the missing side from cos v= -5/6=(adjacent)/(hypote n use)
we use pythagorean theorem to find the length of the opposite side

o=sqrt(6^2-(-5)^2)=sqrt(36-25)=sqrt11. Since, v is in Quadrant III then o=-sqrt11 and sin v=-sqrt11/6

Therefore,

cos(u-v)=cosucosv-sinusinv

=(-4/5)(-5/6)-(3/5)(-sqrt11/6)

=((-2cancel4) /cancel5)(-cancel5/(3cancel6)-(cancel3/5)(-sqrt11/(2cancel6))

=2/3 + sqrt11/10

:.=(20+3sqrt11)/30