Video Explanation

Text Explanation

If you are not familiar with unit circle trigonometry and with radians, I would suggest first watching the lessons on these topic, in the Related Lessons below.  I will give a brief overview here, but for a deeper understanding, it's important to watch those lessons.  

Here's the unit circle, with a ray drawn, an angle  that terminates in the Quadrant III.  

In the unit-circle trigonometry system, all angles are measured counterclockwise from the positive x-axis.  For example, this angle starts at the positive x-axis, and it wraps around to the ray in Q III where the angle ends.  That ray intersects the unit circle at a point we call (x, y), and the sine and cosine functions are defined in terms of the coordinates of this point.  According to the unit circle trig definitions, 

cos() = x

sin() = y

Again, if this 100% new to you, watch those lesson for a little more about how we got from SOHCAHTOA to this!  

The region specified in this question is  ≤  ≤ 2.  An angle of  radians is the same as 180°: this is the angle that starts at the positive x-axis and ends on the negative x-axis, at the point (–1, 0).  Using our unit circle definitions, we can say: 

cos() = –1

sin() = 0 

Angle of 2 radians is the same as 360°: this is the angle that starts at the positive x-axis, wraps all the way around, and winds up back where it started.  If you spin around 360°, then you end up facing the same way you were staring.  This ends at the same place as 0° or 0 radians 

cos(2) = cos(0) = 1

sin(2) = sin(0) = 0

Both of these point, the start and end of the interval in question, are points on the x-axis, points where the sin() = 0.  The points between these are all the values below the x-axis, all the angles in QIII and QIV.  

What we want in this question is an angle in that region where cos() = 0.  Well, cosine is the x-coordinate.  We are interested in a place where x = 0.  Well, x = 0 is the equation of the y-axis, and all the points were x = 0 lie on the y-axis.  Where does the y-axis intersect the unit circle between these points? 

This happens at the very bottom of the unit circle, at the point (0, –1).  The angle there is 270° or  radians.  If  = , then cos() = x = 0, and sin() = y = –1.  

This is the only place in the entire allowed interval where the y-axis crosses the unit circle, so the only value that sin() takes when cos() = 0 in this interval sin() = –1. 

Answer = (A)