Question

identify the ordered pairs on the unit circle corresponding to each real number (t). write your answer as a simplified fraction, if necessary. part 1 of 2 (a) (t =-\frac{14pi}{3}) corresponds to the point ((x,y)=left(-\frac{1}{2},-\frac{sqrt{3}}{2}
ight)). part 1 / 2 part 2 of 2 (b) (t=\frac{pi}{3}) corresponds to the point ((x,y)=square).

Explanation:

Step1: Recall unit - circle properties

For a real number \(t\) on the unit circle, the coordinates of the corresponding point \((x,y)\) are given by \(x = \cos t\) and \(y=\sin t\).

Step2: Calculate \(x\) and \(y\) for \(t=\frac{\pi}{3}\)

When \(t = \frac{\pi}{3}\), we know that \(\cos\frac{\pi}{3}=\frac{1}{2}\) and \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\). So the point \((x,y)=(\frac{1}{2},\frac{\sqrt{3}}{2})\).

Answer:

\((\frac{1}{2},\frac{\sqrt{3}}{2})\)