Find the exact value of each of the remaining trigonometric functions of \( \theta \) \[ \cos \theta=\frac{12}{13}, 270^{\circ}<\theta<360^{\circ} \]

To find the remaining trigonometric functions of \( \theta \) given that \( \cos \theta = \frac{12}{13} \) and \( 270^\circ < \theta < 360^\circ \), we start by determining the sine of \( \theta \). Since \( \theta \) is in the fourth quadrant, the sine function will be negative. We can use the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting the value of \( \cos \theta \): \[ \sin^2 \theta + \left(\frac{12}{13}\right)^2 = 1 \] Calculating \( \left(\frac{12}{13}\right)^2 \): \[ \left(\frac{12}{13}\right)^2 = \frac{144}{169} \] Now substituting this back into the identity: \[ \sin^2 \theta + \frac{144}{169} = 1 \] To isolate \( \sin^2 \theta \), we subtract \( \frac{144}{169} \) from both sides: \[ \sin^2 \theta = 1 - \frac{144}{169} \] Converting 1 to a fraction with a denominator of 169: \[ 1 = \frac{169}{169} \] Thus, \[ \sin^2 \theta = \frac{169}{169} - \frac{144}{169} = \frac{25}{169} \] Taking the square root of both sides gives: \[ \sin \theta = \pm \sqrt{\frac{25}{169}} = \pm \frac{5}{13} \] Since \( \theta \) is in the fourth quadrant, where sine is negative, we have: \[ \sin \theta = -\frac{5}{13} \] Now we can find the remaining trigonometric functions: 1. **Tangent**: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{5}{13}}{\frac{12}{13}} = -\frac{5}{12} \] 2. **Cosecant** (the reciprocal of sine): \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{5}{13}} = -\frac{13}{5} \] 3. **Secant** (the reciprocal of cosine): \[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{12}{13}} = \frac{13}{12} \] 4. **Cotangent** (the reciprocal of tangent): \[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{-\frac{5}{12}} = -\frac{12}{5} \] Now we summarize all the trigonometric functions: - \( \sin \theta = -\frac{5}{13} \) - \( \cos \theta = \frac{12}{13} \) - \( \tan \theta = -\frac{5}{12} \) - \( \csc \theta = -\frac{13}{5} \) - \( \sec \theta = \frac{13}{12} \) - \( \cot \theta = -\frac{12}{5} \) Thus, the exact values of the remaining trigonometric functions of \( \theta \) are: \[ \sin \theta = -\frac{5}{13}, \quad \tan \theta = -\frac{5}{12}, \quad \csc \theta = -\frac{13}{5}, \quad \sec \theta = \frac{13}{12}, \quad \cot \theta = -\frac{12}{5} \]