The radius of the unit circle intersects the circle at the point where the angle is $\frac{\pi }{4}$ radians. What is the approximate value of $y$ at this point?

• A. $0.866$
• B. $1$
• C. $0.707$
• D. $0.5$

Answer: (C)

Verified step by step guidance

1. Recognize that the problem involves a point on the unit circle corresponding to an angle of \(\frac{\pi}{4}\) radians. The unit circle has a radius of 1, and any point on it can be represented as \((x, y) = (\cos \theta, \sin \theta)\) where \(\theta\) is the angle in radians.
2. Identify that the \(y\)-coordinate of the point on the unit circle at angle \(\frac{\pi}{4}\) radians is given by \(y = \sin \left( \frac{\pi}{4} \right)\).
3. Recall the exact value of \(\sin \left( \frac{\pi}{4} \right)\), which is \(\frac{\sqrt{2}}{2}\), a common special angle in trigonometry.
4. Understand that \(\frac{\sqrt{2}}{2}\) is approximately equal to 0.707, which matches the approximate value of \(y\) at this point on the unit circle.
5. Therefore, to find the approximate value of \(y\), calculate or recall the sine of \(\frac{\pi}{4}\) radians, which gives the \(y\)-coordinate of the point where the radius intersects the unit circle.