Given that [tex] \operatorname{sin} \theta = 0.3091 [/tex] and [tex] \theta [/tex] is in quadrant I, find [tex] \operatorname{sin} \frac{\theta}{2} [/tex].

[tex] \operatorname{sin} \frac{\theta}{2} = \square [/tex]

(Round to four decimal places as needed.)



Answer :

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].