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4-113.

Find all values of $θ$ where $θ\ € [0, 2\pi]$ ($θ$ is an element of the interval from $0$ to $2\pi$).

1. $\text{csc }θ =\frac { 2 \sqrt { 3 } } { 3 }$

1. $\text{sec }θ = −\sqrt { 2 }$

1. $\text{tan }θ$ is undefined

1. $\text{cot }θ = -1$

Rewrite $\text{csc}θ$ in terms of $\text{sin}θ$.

Flip the fraction on both sides.

Rationalize the denominator.

Simplify.

The sine function is positive in Quadrants I and II.

$\frac{1}{\text{sin}\theta}=\frac{2\sqrt{3}}{3}$

$\text{sin}\theta=\frac{3}{2\sqrt{3}}$

$\text{sin}\theta=\frac{3}{2\sqrt3}\left(\frac{\sqrt3}{\sqrt3}\right)$

$\text{sin}\theta=\frac{\sqrt3}{2}$

$\text{Therefore, }\theta=\frac{\pi}{3},\frac{2\pi}{3}.$

Use the same thinking process as in (a).

$\frac{\text{sin}\theta}{\text{cos}\theta}$

Use the same thinking process as in (a).