Questions: Let cos θ=2, with 3π<θ<2π. Solve for the remaining trig values: (d) csc θ= (e) cot θ=

Solution Steps

To solve for the remaining trigonometric values given that \(\cos \theta = 2\) and \(3\pi < \theta < 2\pi\), we need to recognize that \(\cos \theta = 2\) is not possible since the range of the cosine function is \([-1, 1]\). Therefore, there must be an error in the given problem. However, if we assume a hypothetical scenario where \(\cos \theta\) could be 2, we would use the definitions of the trigonometric functions to find \(\csc \theta\) and \(\cot \theta\).

1. Recognize that \(\cos \theta = 2\) is not possible within the range of the cosine function.
2. If we hypothetically proceed, we would use the definitions of \(\csc \theta\) and \(\cot \theta\):
   • \(\csc \theta = \frac{1}{\sin \theta}\)
   • \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
3. Since \(\cos \theta = 2\), we would need to find \(\sin \theta\) using the Pythagorean identity, but this would lead to an imaginary number.

Given \(\cos \theta = 2\), we note that this value is outside the range of the cosine function, which is \([-1, 1]\). However, for the sake of the problem, we proceed with the given value.

Using the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] we solve for \(\sin \theta\): \[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - 2^2 = 1 - 4 = -3 \] \[ \sin \theta = \sqrt{-3} = \sqrt{3}i \approx 1.7321i \]

The cosecant function is defined as: \[ \csc \theta = \frac{1}{\sin \theta} \] Substituting the value of \(\sin \theta\): \[ \csc \theta = \frac{1}{1.7321i} \approx -0.5774i \]

The cotangent function is defined as: \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \] Substituting the values of \(\cos \theta\) and \(\sin \theta\): \[ \cot \theta = \frac{2}{1.7321i} \approx -1.1547i \]

Final Answer