Sin alpha=12/13 , p/2<alpha<p

To find the values of \( \cos(\alpha) \) and \( \tan(\alpha) \), given that \( \sin(\alpha) = \frac{12}{13} \) and \( \frac{\pi}{2} < \alpha < \pi \), we first note that this means \( \alpha \) is in the second quadrant.

1. Find \( \cos(\alpha) \): Using the Pythagorean identity: \[ \sin^2(\alpha) + \cos^2(\alpha) = 1 \] Substituting the value of \( \sin(\alpha) \): \[ \left(\frac{12}{13}\right)^2 + \cos^2(\alpha) = 1 \] \[ \frac{144}{169} + \cos^2(\alpha) = 1 \] \[ \cos^2(\alpha) = 1 - \frac{144}{169} = \frac{25}{169} \] \[ \cos(\alpha) = -\sqrt{\frac{25}{169}} = -\frac{5}{13} \] (The cosine is negative in the second quadrant.)

2. Find \( \tan(\alpha) \): The tangent function is given by: \[ \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} \] Substituting the values we have: \[ \tan(\alpha) = \frac{\frac{12}{13}}{-\frac{5}{13}} = -\frac{12}{5} \]

Therefore, the values are:

• \( \cos(\alpha) = -\frac{5}{13} \)
• \( \tan(\alpha) = -\frac{12}{5} \)