Question

question #9 determine the angle or angles on the unit circle that have tanθ = undefined. θ = 0π and θ = π and θ = 2π θ = 0π and θ = 2π θ = π/2 and θ = 3π/2 θ = π/2 question #10 determine the angle or angles on the unit circle that have cosθ = -√2/2. θ = 3π/4 and θ = 5π/4 θ = 0π and θ = 2π θ = 2π/3 and θ = 4π/3 θ = 5π/6 and θ = 7π/6

Explanation:

Step1: Recall tangent formula

Recall that $\tan\theta=\frac{\sin\theta}{\cos\theta}$. Tangent is undefined when $\cos\theta = 0$.

Step2: Find angles on unit - circle

On the unit - circle, $\cos\theta = 0$ when $\theta=\frac{\pi}{2}+k\pi$, $k\in\mathbb{Z}$. In the range $[0, 2\pi]$, the angles are $\theta=\frac{\pi}{2}$ and $\theta=\frac{3\pi}{2}$.

Step1: Recall cosine values on unit - circle

We know that on the unit - circle, the $x$ - coordinate represents $\cos\theta$. We need to find $\theta$ such that $\cos\theta=-\frac{\sqrt{2}}{2}$.

Step2: Identify angles

In the range $[0, 2\pi]$, the angles for which $\cos\theta =-\frac{\sqrt{2}}{2}$ are $\theta=\frac{3\pi}{4}$ and $\theta=\frac{5\pi}{4}$ since the cosine function has a period of $2\pi$ and its values are symmetric about the $y$ - axis in the unit - circle.

Answer:

$\theta=\frac{\pi}{2}$ and $\theta=\frac{3\pi}{2}$