if $sin\thetaapprox - 0.7660$, which of the following represents an approximate value of $\tan\theta$, for…

Explanation:

Step1: Find cosine value

Since $\sin^{2}\theta+\cos^{2}\theta = 1$, then $\cos\theta=\pm\sqrt{1 - \sin^{2}\theta}$. Given $\sin\theta\approx - 0.7660$ and $180^{\circ}<\theta<270^{\circ}$ (in the third - quadrant where cosine is negative), so $\cos\theta=-\sqrt{1-( - 0.7660)^{2}}=-\sqrt{1 - 0.586756}=-\sqrt{0.413244}\approx - 0.6428$.

Step2: Calculate tangent value

The formula for tangent is $\tan\theta=\frac{\sin\theta}{\cos\theta}$. Substitute $\sin\theta\approx - 0.7660$ and $\cos\theta\approx - 0.6428$ into the formula, we get $\tan\theta=\frac{-0.7660}{-0.6428}\approx1.1916$.

Answer:

1.1916