3. given $cos \theta=-\frac{4}{9}$, where $180^{circ} leq \theta leq 270^{circ}$, a) state the other five…

Explanation:

Step1: Recall Pythagorean identity

We know that (\sin^{2}\theta+\cos^{2}\theta = 1). Given (\cos\theta=-\frac{4}{9}), then (\sin^{2}\theta=1 - \cos^{2}\theta). Substitute (\cos\theta) into the formula: (\sin^{2}\theta=1-\left(-\frac{4}{9}\right)^{2}=1-\frac{16}{81}=\frac{81 - 16}{81}=\frac{65}{81}). Since (180^{\circ}\leq\theta\leq270^{\circ}), (\sin\theta<0), so (\sin\theta=-\frac{\sqrt{65}}{9}).

Step2: Calculate (\tan\theta)

The formula for (\tan\theta) is (\tan\theta=\frac{\sin\theta}{\cos\theta}). Substitute (\sin\theta =-\frac{\sqrt{65}}{9}) and (\cos\theta=-\frac{4}{9}) into the formula: (\tan\theta=\frac{-\frac{\sqrt{65}}{9}}{-\frac{4}{9}}=\frac{\sqrt{65}}{4}).

Step3: Calculate (\sec\theta)

The formula for (\sec\theta) is (\sec\theta=\frac{1}{\cos\theta}). Substitute (\cos\theta =-\frac{4}{9}) into the formula: (\sec\theta=-\frac{9}{4}).

Answer:

a) (\sin\theta=-\frac{\sqrt{65}}{9}), (\tan\theta=\frac{\sqrt{65}}{4}), (\sec\theta =-\frac{9}{4}) b) (\theta\approx 195.9^{\circ}) (using (\theta=\cos^{- 1}\left(-\frac{4}{9}\right)+360^{\circ}) and considering the range (180^{\circ}\leq\theta\leq270^{\circ}), (\theta = 180^{\circ}+\cos^{-1}\left(\frac{4}{9}\right)\approx180 + 64.1=195.9^{\circ}))