Question

suppose that θ is an angle in standard position whose terminal side intersects the unit circle at (left( -\frac{3sqrt{5}}{7}, \frac{2}{7}
ight)) find the exact values of (sec\theta), (cos\theta), and (cot\theta).

Explanation:

Step1: Recall unit circle coordinates

For an angle \(\theta\) in standard position with terminal side intersecting the unit circle at \((x,y)\), we have \(x = \cos\theta\) and \(y=\sin\theta\). Here, \(x = -\frac{3\sqrt{5}}{7}\), \(y=\frac{2}{7}\). So \(\cos\theta=-\frac{3\sqrt{5}}{7}\).

Step2: Find \(\sec\theta\)

Recall that \(\sec\theta=\frac{1}{\cos\theta}\). Substituting \(\cos\theta = -\frac{3\sqrt{5}}{7}\), we get \(\sec\theta=\frac{1}{-\frac{3\sqrt{5}}{7}}=-\frac{7}{3\sqrt{5}}\). Rationalizing the denominator (multiply numerator and denominator by \(\sqrt{5}\)): \(\sec\theta = -\frac{7\sqrt{5}}{3\times5}=-\frac{7\sqrt{5}}{15}\).

Step3: Find \(\cot\theta\)

Recall that \(\cot\theta=\frac{\cos\theta}{\sin\theta}\). We know \(\cos\theta = -\frac{3\sqrt{5}}{7}\) and \(\sin\theta=\frac{2}{7}\). So \(\cot\theta=\frac{-\frac{3\sqrt{5}}{7}}{\frac{2}{7}}=-\frac{3\sqrt{5}}{2}\).

Answer:

\(\sec\theta = \boldsymbol{-\frac{7\sqrt{5}}{15}}\), \(\cos\theta = \boldsymbol{-\frac{3\sqrt{5}}{7}}\), \(\cot\theta = \boldsymbol{-\frac{3\sqrt{5}}{2}}\)