Question

suppose an arc of length s lies on the unit circle x² + y² = 1, starting at the point (1,0) and terminating at the point (x,y). use a calculator to find the approximate coordinates for (x,y). s = 0.5

Explanation:

Step1: Recall arc - length formula on unit circle

On the unit circle $x^{2}+y^{2}=1$, if the arc length is $s$ starting from $(1,0)$, and the angle subtended at the center of the circle is $\theta$ (in radians), then $s = \theta$ (since the radius $r = 1$ and $s=r\theta$). Here $s = 0.5$, so $\theta=0.5$ radians.

Step2: Use trigonometric relations

We know that for a point $(x,y)$ on the unit circle, $x=\cos\theta$ and $y = \sin\theta$. Substitute $\theta = 0.5$ into these formulas.
$x=\cos(0.5)\approx0.8776$
$y=\sin(0.5)\approx0.4794$

Answer:

$(0.8776,0.4794)$