) = 0, and
the value of sin(
When cos = 0
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,
) = x
) = 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
. 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:
) = –1
) = 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(0) = 1
) = 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
, 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)
Watch the lessons below for more detailed explanations of the concepts tested in this question. And don't worry, you'll be able to return to this answer from the lesson page.