Question

give the degree measure of \\(\theta\\).\\(\theta = \cot^{-1}(-1)\\)\\(\bigcirc\\) a. \\(90^{\circ}\\)\\(\bigcirc\\) b. \\(180^{\circ}\\)\\(\bigcirc\\) c. \\(135^{\circ}\\)\\(\bigcirc\\) d. \\(45^{\circ}\\)

Explanation:

Step1: Recall the range of \(y = \cot^{-1}x\)

The range of \(y=\cot^{-1}x\) is \((0^{\circ},180^{\circ})\).

Step2: Use the identity \(\cot\theta=\frac{\cos\theta}{\sin\theta}\)

We know that \(\cot\theta=- 1\), so \(\frac{\cos\theta}{\sin\theta}=-1\), which implies \(\cos\theta=-\sin\theta\).
In the range \((0^{\circ},180^{\circ})\), when \(\theta = 135^{\circ}\), \(\cos135^{\circ}=-\frac{\sqrt{2}}{2}\) and \(\sin135^{\circ}=\frac{\sqrt{2}}{2}\), so \(\cot135^{\circ}=\frac{\cos135^{\circ}}{\sin135^{\circ}}=-1\)

Answer:

C. \(135^{\circ}\)