Question

suppose that θ is an angle in standard position whose terminal side intersects the unit circle at \\(\left( -\frac{12}{13}, -\frac{5}{13} \
ight)\\). find the exact values of \\(\cos\theta\\), \\(\tan\theta\\), and \\(\sec\theta\\).

Explanation:

Step1: Recall unit circle coordinates

For a point \((x, y)\) on the unit circle, \(\cos\theta = x\), \(\sin\theta = y\). Here, \(x = -\frac{12}{13}\), \(y = -\frac{5}{13}\). So \(\cos\theta = -\frac{12}{13}\).

Step2: Calculate \(\tan\theta\)

\(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and \(\sin\theta = -\frac{5}{13}\). So \(\tan\theta=\frac{-\frac{5}{13}}{-\frac{12}{13}}=\frac{5}{12}\).

Step3: Calculate \(\sec\theta\)

\(\sec\theta=\frac{1}{\cos\theta}\), so \(\sec\theta=\frac{1}{-\frac{12}{13}}=-\frac{13}{12}\).

Answer:

\(\cos\theta = -\frac{12}{13}\), \(\tan\theta = \frac{5}{12}\), \(\sec\theta = -\frac{13}{12}\)