1. What is the reference angle of -265 degrees?

Question

2. Find the exact value of csc. (480) degrees?
Is the answer -3/2(3.14pi) for number 2?

Answer 1

#1 Add 360 degrees to get 95 degrees. The reference angle is the angle from 95 degrees to the x-axis. 180 degrees minus 95 degrees is 85 degrees, so 85 degrees is your reference angle.

#2 Subtract 360 degrees to get 120 degrees. In radians, this is 2Pi/3. The sin(2Pi/3) is positive since in quadrant II and is the same value as sin(Pi/3) = sqrt(3)/2. Hence csc(2Pi/3)=2/sqrt(3). If you rationalize this, you get 2 sqrt(3) / 3.

Answer 2

The reference angle of -265 degrees: add 360 degrees to get 95 degrees --- this is the coterminal angle. To get the first quadrant related angle...you look at where the angle lies. This angle lies in the second quadrant and is 85 degrees from the x axis. This means the related angled is 85 degrees.

csc 480 = csc 120 = csc 60 This is by CAST rule.

csc is the reciprocal of sin. Therefore since sin 60 is root 3 over 2, csc 480 = 2/ sqrt 3.

Answer 3

The reference angle is made with respect to the x-axis, so -265 is same as +95 degrees wich has a reference angle of 85 degrees.

csc 480 = csc(480-360) = csc(120) = 1/sin(120)=1/sin(90+30)=
1/cos 30 =1/(2/sqrt(3)) = (sqrt (3))/2

Since the angle lies in the 2nd quadrant the sin and its reciprocal the csc are both positive there.