Question

If `secu=-6`, find `sin(u/2),cos(u/2)` and `tan(u/2)`?

Answer 1

See below.

Explanation:

Using following identities:

`color(red)(tanx=sinx/cosx)`

`color(red)(cos^2(x/2)=1/2(1+cos(x)))`

`color(red)(sin^2(x/2)=1/2(1-cos(x)))`

`color(red)(secx=1/cosx)`
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Observe that as `u` is in `Q2`, `u/2` is in `Q1` and all trigonometric ratios of `u/2` are positive.

`sec(u)=1/cos(u)`

`:.`

`1/cos(u)=-6` , `cos(u)=1/-6`

`color(green)(cos(u/2))=sqrt(1/2(1+cos(u)))=sqrt(1/2(1-1/6))=color(blue)(sqrt(5/12))`

`color(green)(sin(u/2))=sqrt(1/2(1-cos(u)))=sqrt(1/2(1-(-1/6)))=color(blue)(sqrt(7/12))`

`color(green)(tan(u/2))=sin(u/2)/cos(u/2)=(sqrt(7/12))/(sqrt(5/12))=color(blue)((sqrt(35))/5)`