How do you find the exact values of $sin (\frac{u}{2}), cos (\frac{u}{2}), tan (\frac{u}{2})$ $sin(u/2), cos(u/2), tan(u/2)$ using the half angle formulas given $sec u = - \frac{7}{2}, \frac{π}{2} < u < π$ $secu=-7/2, pi/2<u<pi$ ?

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Answer 1 · 2018-02-25T13:50:17

See below.

Identities:

$sec x = \frac{1}{cos x}$ $color(red)bb(secx=1/cosx)$

$\frac{sin (\frac{x}{2})}{cos (\frac{x}{2})} = tan (\frac{x}{2})$ $color(red)bb(sin(x/2)/cos(x/2)=tan(x/2))$

$cos (\frac{x}{2}) = \sqrt{\frac{1}{2} (1 + cos x)}$ $color(red)bb(cos(x/2)=sqrt(1/2(1+cosx)))$

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

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

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

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

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

Note the signs:

If $u$ is in quadrant II

Then:

$u/2$ is in quadrant I

How do you find the exact values of sin(u/2), cos(u/2), tan(u/2)sin(u2),cos(u2),tan(u2)sin(u/2), cos(u/2), tan(u/2) using the half angle formulas given secu=-7/2, pi/2<u<pisecu=−72,π2<u<πsecu=-7/2, pi/2<u<pi?