Question

How do you find the exact values of `sin(u/2), cos(u/2), tan(u/2)` using the half angle formulas given `secu=-7/2, pi/2<u<pi`?

Answer 1

See below.

Explanation:

Identities:

`color(red)bb(secx=1/cosx)`

`color(red)bb(sin(x/2)/cos(x/2)=tan(x/2))`

`color(red)bb(cos(x/2)=sqrt(1/2(1+cosx)))`

`color(red)bb(sin(x/2)=sqrt(1/2(1-cosx)))`

`sec(u)=-7/2=>1/cos(u)=-7/2=>cos(u)=-2/7`

`sin(u/2)=sqrt(1/2(1-(-2/7)))=sqrt((9/14))=3/sqrt(14)=color(blue)((3sqrt(14))/14)`

`cos(u/2)=sqrt(1/2(1+(-2/7)))=sqrt(5/14)=color(blue)((sqrt(70))/14)`

`tan(u/2)=((3sqrt(14))/14)/(-(sqrt(70))/14)=(3sqrt(14))/(sqrt(70))=color(blue)((3sqrt(5))/5)`

Note the signs:

If `u` is in quadrant II

Then:

`u/2` is in quadrant I