Question

points a, b, and c are plotted on the unit circle. which of the coordinates is equal to sin(310°)?

Explanation:

Step1: Recall unit circle definition

On the unit circle, for an angle \(\theta\) in standard position, the coordinates of the point where the terminal side intersects the unit circle are \((\cos\theta, \sin\theta)\). So, \(\sin\theta\) is the \(y\)-coordinate of the point corresponding to angle \(\theta\).

Step2: Analyze the angle \(310^\circ\)

First, find the reference angle for \(310^\circ\). Since \(310^\circ = 360^\circ - 50^\circ\), it is in the fourth quadrant. In the fourth quadrant, sine values (the \(y\)-coordinates) are negative.

Now, let's analyze the points:

• Point \(A\) is in the first quadrant, so its \(y\)-coordinate is positive.
• Point \(B\) is in the third quadrant (wait, no, looking at the graph, \(B\) is on the left - lower part, but actually, \(310^\circ\) is in the fourth quadrant. Wait, let's re - check the quadrants. The fourth quadrant is where \(x>0\) and \(y < 0\). Point \(C\) is in the fourth quadrant (since it's in the lower - right part of the unit circle), point \(B\) is in the third quadrant (lower - left), point \(A\) is in the first quadrant (upper - right).

Since \(\sin(310^\circ)\) is the \(y\)-coordinate of the point on the unit circle corresponding to \(310^\circ\), and \(310^\circ\) is in the fourth quadrant (where \(y\) is negative), we need to find the point with a negative \(y\)-coordinate. Among \(A\), \(B\), and \(C\), point \(C\) is in the fourth quadrant, so its \(y\)-coordinate is negative, and point \(B\) is in the third quadrant ( \(x\) negative, \(y\) negative), but \(310^\circ\) is \(360 - 50\), so the reference angle is \(50^\circ\), and the coordinates should be \((\cos(310^\circ),\sin(310^\circ))=(\cos(50^\circ),-\sin(50^\circ))\), so the \(y\)-coordinate is negative and the \(x\)-coordinate is positive. So the point with positive \(x\) and negative \(y\) is \(C\). So the \(y\)-coordinate of point \(C\) is equal to \(\sin(310^\circ)\).

Answer:

The \(y\)-coordinate of point \(C\) (i.e., the \(y\)-value of point \(C\) on the unit circle) is equal to \(\sin(310^\circ)\).