Find the exact value of each of the remaining trigonometric functions of \( \theta \) \[ \cos \theta=\frac{4}{5}, 270^{\circ}<\theta<360^{\circ} \]

1. Since \( \cos \theta = \frac{4}{5} \) and \( 270^\circ < \theta < 360^\circ \) (Quadrant IV), we know: - \( \cos \theta > 0 \) - \( \sin \theta < 0 \) 2. Use the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substitute \( \cos \theta = \frac{4}{5} \): \[ \sin^2 \theta + \left(\frac{4}{5}\right)^2 = 1 \quad \Rightarrow \quad \sin^2 \theta + \frac{16}{25} = 1 \] Solve for \( \sin^2 \theta \): \[ \sin^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \] Taking the square root (and choosing the negative value in Quadrant IV): \[ \sin \theta = -\frac{3}{5} \] 3. Compute the remaining trigonometric functions: - Tangent: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{3}{5}}{\frac{4}{5}} = -\frac{3}{4} \] - Secant (reciprocal of cosine): \[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{4}{5}} = \frac{5}{4} \] - Cosecant (reciprocal of sine): \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3} \] - Cotangent (reciprocal of tangent): \[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{-\frac{3}{4}} = -\frac{4}{3} \] 4. The exact values of the remaining trigonometric functions are: - \(\sin \theta = -\frac{3}{5}\) - \(\tan \theta = -\frac{3}{4}\) - \(\sec \theta = \frac{5}{4}\) - \(\csc \theta = -\frac{5}{3}\) - \(\cot \theta = -\frac{4}{3}\)