How do you use half-angle identities to find the exact values of sin (u/2) & cos (u/2) when sin(u)= - 4/5 and 3pi/2 <u<2pi?

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How do you use half-angle identities to find the exact values of sin (u/2) & cos (u/2) when sin(u)= - 4/5 and 3pi/2 <u<2pi?

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Answer 1 · 2016-01-25T02:08:25

$cos (\frac{u}{2}) = - 2 \frac{\sqrt{5}}{5}$ $cos (u/2) = -2sqrt5/5$
$sin (\frac{u}{2}) = \frac{\sqrt{5}}{5}$ $sin (u/2) = sqrt5/5$

Explanation:

$sin u = - \frac{4}{5}$ $sin u = - 4/5$ --> ${cos}^{2} u = 1 - {sin}^{2} u = 1 - \frac{16}{25} = \frac{9}{25}$ $cos ^2 u = 1 - sin^2 u = 1 - 16/25 = 9/25$
$cos u = \frac{3}{5}$ $cos u = 3/5$ (cos u positive since u is in the Quadrant IV).
To find $sin (\frac{u}{2})$ $sin (u/2)$ and $cos (\frac{u}{2})$ $cos (u/2)$ apply the trig identities:
$cos 2 a = 2 {cos}^{2} a - 1$ $cos 2a = 2cos^2 a - 1$
$cos 2 a = 1 - 2 {sin}^{2} a$ $cos 2a = 1 - 2sin^2 a$
a. Find $cos (\frac{u}{2})$ $cos (u/2)$
$cos u = \frac{3}{5} = 2 {cos}^{2} (\frac{u}{2}) - 1$ $cos u = 3/5 = 2cos^2 (u/2) - 1$
$2 {cos}^{2} (\frac{u}{2}) = 1 + \frac{3}{5} = \frac{8}{5}$ $2cos^2 (u/2) = 1 + 3/5 = 8/5$
${cos}^{2} (\frac{u}{2}) = \frac{8}{10}$ $cos^2 (u/2) = 8/10$
$cos (\frac{u}{2}) = \frac{\sqrt{8}}{\sqrt{10}} = \pm \frac{\sqrt{80}}{10} = \pm 4 \frac{\sqrt{5}}{10} = \pm 2 \frac{\sqrt{5}}{5}$ $cos (u/2) = sqrt8/sqrt10 = +- sqrt80/10 = +- 4sqrt5/10 = +- 2sqrt5/5$
$cos (\frac{u}{2}) = - 2 \frac{\sqrt{5}}{5}$ $cos (u/2) = - 2sqrt5/5$ ( $cos (\frac{u}{2})$ $cos (u/2)$ negative, Quadrant II)
b. Find $sin (\frac{u}{2})$ $sin (u/2)$
$\frac{3}{5} = 1 - 2 {sin}^{2} (\frac{u}{2})$ $3/5 = 1 - 2sin^2 (u/2)$
$2 {sin}^{2} (\frac{u}{2}) = 1 - \frac{3}{5} = \frac{2}{5}$ $2sin^2 (u/2) = 1 - 3/5 = 2/5$
${sin}^{2} (\frac{u}{2}) = \frac{2}{10} = \frac{1}{5}$ $sin^2 (u/2) = 2/10 = 1/5$
$sin (\frac{u}{2}) = \pm \frac{1}{\sqrt{5}} = \pm \frac{\sqrt{5}}{5}$ $sin (u/2) = +- 1/sqrt5 = +- sqrt5/5$
$sin (\frac{u}{2}) = \frac{\sqrt{5}}{5}$ $sin (u/2) = sqrt5/5$ ( $sin (\frac{u}{2})$ $sin (u/2)$ positive, Quadrant II)

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How do you use half-angle identities to find the exact values of sin (u/2) & cos (u/2) when sin(u)= - 4/5 and 3pi/2 <u<2pi?

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