CPM Homework Help : PC Problem 4-113

4-113.

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

Step 1(a):

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

Step 2(a):

Flip the fraction on both sides.

Step 3(a):

Rationalize the denominator.

Step 4(a):

Simplify.

Step 5(a):

The sine function is positive in Quadrants I and II.

Answer 1(a):

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

Answer 2(a):

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

Answer 3(a):

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

Answer 4(a):

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

Answer 5(a):

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

Hint (b):

Use the same thinking process as in (a).

Hint (c):

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

Hint (d):

Use the same thinking process as in (a).