Question

Express csc(-330) as a trigonometric function of an angle in Quadrant I

If anyone could please explain how to get to the answer that would be great!! I am so lost

Answer 1

`\csc(30\degree)`

Explanation:

The given trigonometric expression is

`\csc(-330\degree)`

Recall that;

`\csc(\theta)=(1)/(\sin(\theta))`

This implies that;

`\csc(-330\degree)=(1)/(\sin(-330))`

Recall again that;

`\sin(-\theta)=-\sin(\theta)`

We apply this property to get;

`\csc(-330\degree)=-(1)/(\sin(330))`

Finally, we apply the following property of the sine function to get;

`\sin(360\degree-\theta)=-\sin(\theta)`

`\csc(-330\degree)=-(1)/(\sin(360\degree-30\degree))`

`\csc(-330\degree)=-(1)/(-\sin(30\degree))`

`\Rightarrow \csc(-330\degree)=(1)/(\sin(30\degree))`

We rewrite using reciprocal ratios to get;

`\Rightarrow \csc(-330\degree)=\csc(30)`