How do you evaluate csc(13pi/3)?

Question

The answer is 2rad3/3.

I only got up to a few steps...but them i got confused.....so first I converted this to degrees by timing it by 180/pi. Then I got the number 780...and I don't know what to do next cause I'm not used to dealing with that big of a number....do u have to find a reference angle or something?

Answer 1

The reference angle is pi/3 (60 degrees) - get it buy subtracting 2pi (360) until you get an acute angle.

csc is the reciprocal of sin.

sin of pi/3 (sin 60) is rad3/2 so the csc value is 2/rad 3 which simplifies to 2 rad 3 / 3.

Answer 2

The conversion to 780 degrees is correct. If you consider the fact that if you were to spin 360 degrees in the left or right direction you end up where you started, you can see that it can be added or subtracted in any multiple. So we have 780 - 360 = 420 degrees. This is still not a "normal" angle, so we must get it closer to 0 by subtracting 360 again. Everry time we subtract 360, it is like turning around and ending up in the exact same spot. We have 420 - 360 = 60 degrees. So if we consider what we have, the angle 780 degrees is made up of 360 + 360 + 60 degrees, or 2 revolutions and a 60 degree turn. Now that we have an equivalent angle (not a refereence angle), we can take the csc. csc is 1/sin, so because the sin 60 = sqrt(3)/2, csc 60 = 2/sqrt(3) or rationalized 2sqrt(3)/3

Answer 3

The number 780 is not in the unit circle, because a circle is 360 degrees. So you subtract 780 - 360 = 420 and you have to subtract is again because 420 is still bigger than 360. 420 - 360 = 60 or pi/3 (radians)

csc is the recepricol of sin, so first take the sin of 60 or pi/3

sin(pi/3) = radical 3 over 2

now you take the recepricol of rad 3 over 2 to get the actualy answer

when you take the recepricol, you'll get 2 over rad 3

but since it's not acceptable to have radicals in the denominator, you have to multiple the denominator and numerator of the fraction by rad 3

you will get 2 rad 3 over 3.