Questions: Find the exact value of each of the remaining trigonometric functions of (theta). (cos theta=-frac56, quad 90^circ<theta<180^circ) (sin theta=fracsqrt116) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression) (tan theta=)

Solution Steps

1. Given \(\cos \theta = -\frac{5}{6}\) and \(90^\circ < \theta < 180^\circ\), we know that \(\theta\) is in the second quadrant where sine is positive and cosine is negative.
2. Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to verify \(\sin \theta\).
3. Calculate \(\tan \theta\) using the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
4. Calculate \(\cot \theta\) as the reciprocal of \(\tan \theta\).
5. Calculate \(\sec \theta\) as the reciprocal of \(\cos \theta\).
6. Calculate \(\csc \theta\) as the reciprocal of \(\sin \theta\).

Using the values of \(\sin \theta\) and \(\cos \theta\), we find: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{11}}{6}}{-\frac{5}{6}} = -\frac{\sqrt{11}}{5} \approx -0.6633 \]

The cotangent is the reciprocal of the tangent: \[ \cot \theta = \frac{1}{\tan \theta} = -\frac{5}{\sqrt{11}} \approx -1.508 \]

The secant is the reciprocal of the cosine: \[ \sec \theta = \frac{1}{\cos \theta} = -\frac{6}{5} = -1.2 \]

The cosecant is the reciprocal of the sine: \[ \csc \theta = \frac{1}{\sin \theta} = \frac{6}{\sqrt{11}} \approx 1.8091 \]

Final Answer