To find [tex]\(\sin\left(\frac{\theta}{2}\right)\)[/tex] given that [tex]\(\sin(\theta) = 0.3091\)[/tex] and [tex]\(\theta\)[/tex] is in the first quadrant, follow these steps:
1. Use the Pythagorean Identity:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
Given [tex]\(\sin(\theta) = 0.3091\)[/tex], we can find [tex]\(\cos(\theta)\)[/tex].
2. Calculate [tex]\(\cos(\theta)\)[/tex]:
First, find [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[
\sin^2(\theta) = (0.3091)^2 = 0.09554481
\][/tex]
Now, using the Pythagorean identity:
[tex]\[
\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - 0.09554481 = 0.90445519
\][/tex]
Since [tex]\(\theta\)[/tex] is in the first quadrant, where both sine and cosine are positive:
[tex]\[
\cos(\theta) = \sqrt{0.90445519} \approx 0.9510
\][/tex]
3. Use the Half-Angle Formula:
The half-angle formula for sine is:
[tex]\[
\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}
\][/tex]
Since [tex]\(\frac{\theta}{2}\)[/tex] is half of an angle in the first quadrant (which keeps it in the first quadrant too), [tex]\(\sin\left(\frac{\theta}{2}\right)\)[/tex] is positive.
4. Substitute [tex]\(\cos(\theta)\)[/tex] into the Half-Angle Formula:
[tex]\[
\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - 0.9510}{2}}
\][/tex]
Calculate the numerator:
[tex]\[
1 - 0.9510 = 0.049
\][/tex]
Calculate the entire expression:
[tex]\[
\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{0.049}{2}} = \sqrt{0.0245}
\][/tex]
[tex]\[
\sin\left(\frac{\theta}{2}\right) \approx 0.15647756687797398
\][/tex]
5. Round the Result:
Round the result to four decimal places:
[tex]\[
\sin\left(\frac{\theta}{2}\right) \approx 0.1565
\][/tex]
Therefore,
[tex]\[
\sin\left(\frac{\theta}{2}\right) \approx 0.1565
\][/tex]
So the value of [tex]\(\sin\left(\frac{\theta}{2}\right)\)[/tex] rounded to four decimal places is [tex]\(0.1565\)[/tex].
