Given that [tex]\(\sin \theta = 0.3626\)[/tex] and [tex]\(\theta\)[/tex] is in the first quadrant, we want to find [tex]\(\sin 2\theta\)[/tex].
To calculate [tex]\(\sin 2\theta\)[/tex], we can use the double-angle identity for sine:
[tex]\[
\sin 2\theta = 2 \sin \theta \cos \theta
\][/tex]
First, since [tex]\(\theta\)[/tex] is in the first quadrant, both [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] are positive. We need to determine [tex]\(\cos \theta\)[/tex] using the Pythagorean identity:
[tex]\[
\cos \theta = \sqrt{1 - \sin^2 \theta}
\][/tex]
Substituting [tex]\(\sin \theta = 0.3626\)[/tex] into the identity, we get:
[tex]\[
\cos \theta = \sqrt{1 - (0.3626)^2}
\][/tex]
Calculating the square of [tex]\(\sin \theta\)[/tex]:
[tex]\[
(0.3626)^2 = 0.13147576
\][/tex]
Next, subtract this result from 1:
[tex]\[
1 - 0.13147576 = 0.86852424
\][/tex]
Now, take the square root:
[tex]\[
\cos \theta = \sqrt{0.86852424} \approx 0.9319448696140775
\][/tex]
With [tex]\(\sin \theta = 0.3626\)[/tex] and [tex]\(\cos θ \approx 0.9319448696140775\)[/tex], we can now find [tex]\(\sin 2\theta\)[/tex]:
[tex]\[
\sin 2\theta = 2 \times 0.3626 \times 0.9319448696140775
\][/tex]
Calculating the product:
[tex]\[
\sin 2\theta \approx 2 \times 0.3626 \times 0.9319448696140775 \approx 0.6758464194441289
\][/tex]
Finally, we round the result to four decimal places:
[tex]\[
\sin 2\theta \approx 0.6758
\][/tex]
Therefore, [tex]\(\sin 2\theta = 0.6758\)[/tex].
