Problem 13 - Precalculus: A Concise Course

Question

Find the point \((x, y)\) on the unit circle that corresponds to the real number \(t\). $$ t=\frac{5 \pi}{6} $$

Step-by-Step Solution

Answer

The point \((- \frac{1}{2}, \frac{\sqrt{3}}{2})\) on the unit circle corresponds to the real number \(t =\frac{5 \pi}{6}\).

Step 1: Understand the Unit Circle

In a unit circle, the 'x' coordinate is the cosine of the angle in radians and the 'y' coordinate is the sine of the angle in radians. As the problem provides an angle \(t = \frac{5 \pi}{6}\), use this to find the coordinates (\(x,y\)).

Step 2: Find x-coordinate

Calculate the cosine of the given angle, \(t\), to find the x-coordinate. So, \(x = \cos(t) = \cos(\frac{5 \pi}{6})\).

Step 3: Find y-coordinate

Calculate the sine of the angle \(t\) to find the y-coordinate. So, \(y = \sin(t) = \sin(\frac{5 \pi}{6})\).

Step 4: Establish final points

The final points \((x, y)\) correspond to the real number \(t\) on the unit circle are \((- \frac{1}{2}, \frac{\sqrt{3}}{2})\).