How do you find cos(x/2) given sin(x)=1/4?

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How do you find cos(x/2) given sin(x)=1/4?

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Answer 1 · 2017-10-17T03:45:01

$cos (\frac{x}{2}) = 0.99$ $cos (x/2) = 0.99$
$cos (\frac{x}{2}) = 0.127$ $cos (x/2) = 0.127$

Explanation:

$sin x = \frac{1}{4}$ $sin x = 1/4$ . Find cos x
${cos}^{2} x = 1 - {sin}^{2} x = 1 - \frac{1}{16} = \frac{15}{16}$ $cos^2 x = 1 - sin^2 x = 1 - 1/16 = 15/16$
$cos x = \pm \frac{\sqrt{15}}{4}$ $cos x = +- sqrt15/4$
There are 2 values of cos x because if $sin x = \frac{1}{4}$ $sin x = 1/4$ , x could either be in Quadrant 1 or in Quadrant 2
Use trig identity:
$2 {cos}^{2} (\frac{x}{2}) = 1 - cos 2 a$ $2cos^2 (x/2) = 1 - cos 2a$
In this case:
${cos}^{2} (\frac{x}{2}) = \frac{1}{2} \pm \frac{\sqrt{15}}{8} = \frac{1}{2} \pm 0.484$ $cos^2 (x/2) = 1/2 +- sqrt15/8 = 1/2 +- 0.484$
a. ${cos}^{2} (\frac{x}{2}) = 0.984$ $cos^2 (x/2) = 0.984$
b. ${cos}^{2} (\frac{x}{2}) = 0.016$ $cos^2 (x/2) = 0.016$
a. $cos (\frac{x}{2}) = \sqrt{0.984} = 0.99$ $cos (x/2) = sqrt(0.984) = 0.99$
b. $cos (\frac{x}{2}) = \sqrt{0.016} = 0.127$ $cos (x/2) = sqrt(0.016) = 0.127$
Check with calculator.
a. $cos (\frac{x}{2}) = 0.99$ $cos (x/2) = 0.99$ --> $\frac{x}{2} = 7^{\circ} 27$ $x/2 = 7^@27$ --> $x = 14^{\circ} 53$ $x = 14^@53$
$sin (14^{\circ} 52) = 0.25$ $sin (14^@52) = 0.25$ . Proved
b. $cos (\frac{x}{2}) = 0.127$ $cos (x/2) = 0.127$ --> $\frac{x}{2} = 82^{\circ} 70$ $x/2 = 82^@70$ --> $x = 165^{\circ} 41$ $x = 165^@41$
$sin (165^{\circ} 41) = 0.25$ $sin (165^@41) = 0.25$ . Proved

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How do you find cos(x/2) given sin(x)=1/4?

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