Question

1. find all angles θ in the interval 0,2π) that satisfy the expression tanθ=undefined.

Explanation:

Step1: Recall the domain of the tangent function

The tangent function \(y = \tan\theta=\frac{\sin\theta}{\cos\theta}\). The function \(y = \tan\theta\) is undefined when \(\cos\theta = 0\).

Step2: Solve the equation \(\cos\theta=0\)

We know that the general solution of the equation \(\cos\theta = 0\) is \(\theta=\frac{\pi}{2}+k\pi\), where \(k\in\mathbb{Z}\).

Step3: Find the solutions in the interval \([0,2\pi]\)

When \(k = 0\), \(\theta=\frac{\pi}{2}\); when \(k = 1\), \(\theta=\frac{\pi}{2}+\pi=\frac{3\pi}{2}\)

Answer:

\(\theta=\frac{\pi}{2},\frac{3\pi}{2}\)