Let's go through the process to find the approximate values of [tex]\(\sin \theta\)[/tex] and [tex]\(\tan \theta\)[/tex] given that [tex]\(\cos \theta \approx 0.3090\)[/tex].
1. Determine [tex]\(\sin \theta\)[/tex] using the Pythagorean identity:
The Pythagorean identity states:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Given [tex]\(\cos \theta \approx 0.3090\)[/tex], we can rearrange the identity to solve for [tex]\(\sin \theta\)[/tex]:
[tex]\[
\sin^2 \theta = 1 - \cos^2 \theta
\][/tex]
Substituting [tex]\(\cos \theta \approx 0.3090\)[/tex]:
[tex]\[
\sin^2 \theta = 1 - (0.3090)^2
\][/tex]
[tex]\[
\sin^2 \theta = 1 - 0.0954
\][/tex]
[tex]\[
\sin^2 \theta = 0.9046
\][/tex]
To find [tex]\(\sin \theta\)[/tex], take the square root of both sides:
[tex]\[
\sin \theta \approx \sqrt{0.9046} \approx 0.9511
\][/tex]
2. Determine [tex]\(\tan \theta\)[/tex] using the definition:
The definition of the tangent function is:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
Substituting [tex]\(\sin \theta \approx 0.9511\)[/tex] and [tex]\(\cos \theta \approx 0.3090\)[/tex]:
[tex]\[
\tan \theta \approx \frac{0.9511}{0.3090} \approx 3.0779
\][/tex]
Therefore, the approximate values of [tex]\(\sin \theta\)[/tex] and [tex]\(\tan \theta\)[/tex] are:
[tex]\[
\sin \theta \approx 0.9511
\][/tex]
[tex]\[
\tan \theta \approx 3.0780
\][/tex]
The correct answer from the given options is:
[tex]\[
\sin \theta \approx 0.9511 ; \tan \theta \approx 3.0780
\][/tex]
