Question

Find the exact value of csc(-1860).

Answer 1

Now, there are 360° in a circle, how many times does 360° go into 1860°?

well, let's check that,   $\bf \cfrac{1860}{360}\implies \cfrac{31}{6}\implies 5\frac{1}{6}\implies 5+\frac{1}{6}$

now, this is a negative angle, so it's going clockwise, like a clock moves, so it goes around the circle clockwise 5 times fully, and then it goes 1/6 extra.

well, we know 360° is in a circle, how many degrees in 1/6 of 360°?  well, is just 360/6 or their product, and that's just 60°.

so -1860, is an angle that goes clockwise, negative, 5 times fully, then goes an extra 60° passed.

5 times fully will land you back at the 0 location, if you move further down 60° clockwise, that'll land you on the IV quadrant, with an angle of -60°.

therefore, the csc(-1860°) is the same as the angle of csc(-60°), which is the same as the csc(360° - 60°) or csc(300°).

$\bf csc(300^o)\implies \cfrac{1}{sin(300^o)}\implies \cfrac{1}{-\frac{\sqrt{3}}{2}}\implies -\cfrac{2}{\sqrt{3}} \\\\\\ \textit{and if we rationalize the denominator}\qquad -\cfrac{2\sqrt{3}}{3}$