To find the approximate value of [tex]\(\tan \theta\)[/tex] for [tex]\(180^{\circ} < \theta < 270^{\circ}\)[/tex] given that [tex]\(\sin \theta \leq -0.7660\)[/tex], we need to follow these steps:
1. Understand the given interval:
- In the interval [tex]\(180^{\circ} < \theta < 270^{\circ}\)[/tex], both [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] are negative, making [tex]\(\tan \theta\)[/tex] positive because [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex].
2. Given value:
- We are given [tex]\(\sin \theta \leq -0.7660\)[/tex]. For our calculations, we'll use [tex]\(\sin \theta = -0.7660\)[/tex].
3. Calculate [tex]\(\cos \theta\)[/tex]:
- We use the Pythagorean identity: [tex]\(\sin^2(\theta) + \cos^2(\theta) = 1\)[/tex].
- Plugging in the value, we get:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
[tex]\[
(-0.7660)^2 + \cos^2(\theta) = 1
\][/tex]
- Squaring [tex]\(-0.7660\)[/tex]:
[tex]\[
0.7660^2 = 0.5876359999999999
\][/tex]
- So,
[tex]\[
0.5876359999999999 + \cos^2(\theta) = 1
\][/tex]
[tex]\[
\cos^2(\theta) = 1 - 0.5876359999999999
\][/tex]
[tex]\[
\cos^2(\theta) = 0.41236400000000006
\][/tex]
- Taking the square root to find [tex]\(\cos(\theta)\)[/tex], and considering [tex]\(\cos \theta\)[/tex] is negative in this interval:
[tex]\[
\cos(\theta) = -\sqrt{0.41236400000000006}
\][/tex]
[tex]\[
\cos(\theta) \approx -0.6428405712149786
\][/tex]
4. Calculate [tex]\(\tan \theta\)[/tex]:
- Using [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex]:
[tex]\[
\tan(\theta) = \frac{-0.7660}{-0.6428405712149786}
\][/tex]
[tex]\[
\tan(\theta) \approx 1.1915862723975996
\][/tex]
5. Conclusion:
- The approximate value of [tex]\(\tan \theta\)[/tex] is around 1.1916.
So, the correct answer is:
[tex]\[
\boxed{1.1916}
\][/tex]
