Recall the definition of the cosecant function in terms of the sine function:
[tex]\csc (x)=\frac{1}{\sin (x)}[/tex]
Draw the angle -210º in the unit circle to find a reference triangle for that angle:
Notice that a 30-60-90 triangle is formed:
Since the side opposite to the 30º angle has a length of 1/2, then:
[tex]\sin (30º)=\frac{1}{2}[/tex]
On the other hand, notice that since the angle of -210º lies in the second quadrant, then the value of sin(-210º) is positive:
[tex]\sin (-210º)=\sin (30º)=\frac{1}{2}[/tex]
Use the definition of the cosecant function to find csc(-210º):
[tex]\csc (-210)=\frac{1}{\sin (-210º)}=\frac{1}{(\frac{1}{2})}=2[/tex]
Therefore, the value of csc(-210º) is equal to 2.
