Question

In the xy-plane, the terminal ray of an angle in standard position intersects with the unit circle at the point with coordinates (a,b). The terminal ray of a second angle in standard position intersects with the unit circle at the point with coordinates (c,d). If the measure of the second angle is twice the measure of the first angle, what are the coordinates c and d, in terms of a and b?

Answer 1

To solve this problem, let's denote the angle corresponding to the point \((a, b)\) on the unit circle as \(\theta\). Thus, we have:

\[
a = \cos(\theta) \quad \text{and} \quad b = \sin(\theta)
\]

For the second angle, which we denote as \(2\theta\), the coordinates of the point where the terminal ray of this angle intersects the unit circle can be expressed using the double angle formulas for sine and cosine. Specifically, the coordinates are given by:

\[
c = \cos(2\theta) \quad \text{and} \quad d = \sin(2\theta)
\]

Using the double angle formulas, we have:

\[
\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)
\]
\[
\sin(2\theta) = 2 \sin(\theta) \cos(\theta)
\]

Substituting the expressions for \(a\) and \(b\) into these formulas, we have:

1. For \(c\):

\[
c = \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = a^2 - b^2
\]

2. For \(d\):

\[
d = \sin(2\theta) = 2 \sin(\theta) \cos(\theta) = 2ab
\]

Thus, the coordinates \( (c, d) \) in terms of \( (a, b) \) are:

\[
c = a^2 - b^2
\]
\[
d = 2ab
\]

In summary, the coordinates of the point where the terminal ray of the angle \(2\theta\) intersects the unit circle, in terms of the coordinates \( (a, b) \) of the angle \(\theta\), are:

\[
(c, d) = (a^2 - b^2, 2ab)
\]