To determine the correct approximate values for [tex]\(\sin \theta\)[/tex] and [tex]\(\tan \theta\)[/tex] given [tex]\(\cos \theta \approx 0.3090\)[/tex] for [tex]\(0^{\circ}<\theta<90^{\circ}\)[/tex], follow these steps:
1. Calculate [tex]\(\sin \theta\)[/tex]:
We use the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Thus,
[tex]\[
\sin^2 \theta = 1 - \cos^2 \theta
\][/tex]
Given [tex]\(\cos \theta \approx 0.3090\)[/tex], we find:
[tex]\[
\cos^2 \theta \approx 0.3090^2 = 0.095481
\][/tex]
Therefore,
[tex]\[
\sin^2 \theta \approx 1 - 0.095481 = 0.904519
\][/tex]
Taking the positive square root (since [tex]\(\theta\)[/tex] is in the first quadrant),
[tex]\[
\sin \theta \approx \sqrt{0.904519} \approx 0.9511
\][/tex]
2. Calculate [tex]\(\tan \theta\)[/tex]:
The tangent function is defined as the ratio of the sine and cosine functions:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
Using the values we have,
[tex]\[
\tan \theta \approx \frac{0.9511}{0.3090} \approx 3.0780
\][/tex]
3. Compare with given options:
Now, we check the given options to see which closely match our calculated values:
- [tex]\(\sin \theta \approx 0.9511, \tan \theta \approx 0.3249\)[/tex]
- [tex]\(\sin \theta \approx 0.9511; \tan \theta \approx 3.0780\)[/tex]
- [tex]\(\sin \theta \approx 3.2362; \tan \theta \approx 0.0955\)[/tex]
- [tex]\(\sin \theta \approx 3.2362; \tan \theta \approx 10.4731\)[/tex]
The values that match our calculations ([tex]\(\sin \theta \approx 0.9511\)[/tex] and [tex]\(\tan \theta \approx 3.0780\)[/tex]) are provided in the second option.
Therefore, the correct approximate values are:
[tex]\[
\sin \theta \approx 0.9511; \tan \theta \approx 3.0780
\][/tex]
