Given a point $(a, b)$ on the unit circle, find the angle $ heta$ (in radians) between the line connecting the origin to the point and the positive x-axis.
Answer 1
Abigail Nelson
To find the angle $\theta$ between the line connecting the origin to the point $(a, b)$ on the unit circle and the positive x-axis, we use the definition of sine and cosine.
Since $(a, b)$ is on the unit circle, we know that $a = \cos(\theta)$ and $b = \sin(\theta)$.
Therefore, $\theta = \arctan \left( \frac{b}{a} \right)$ if $a > 0$.
If $a < 0$, $\theta = \pi + \arctan \left( \frac{b}{a} \right)$.
If $a = 0$ and $b > 0$, $\theta = \frac{\pi}{2}$.
If $a = 0$ and $b < 0$, $\theta = \frac{3\pi}{2}$.
Answer: $\theta = \arctan \left( \frac{b}{a} \right)$ or other corresponding values based on the quadrant.
Answer 2
Ava Martin
Given the coordinates $(a, b)$ on the unit circle, we use trigonometric identities. First, we note that $a^2 + b^2 = 1$.
The angle $ heta$ can be found using the inverse trigonometric functions:
If $a > 0$, $ heta = arccos(a)$.
If $a < 0$:
If $b geq 0$, $ heta = pi – arccos(-a)$.
If $b < 0$, $ heta = pi + arccos(-a)$.
If $a = 0$, we use the values of $b$:
If $b > 0$, $ heta = frac{pi}{2}$.
If $b < 0$, $ heta = frac{3pi}{2}$.
Answer: $ heta = arccos(a)$ or other corresponding values based on the quadrant.
Answer 3
Emma Johnson
For any point $(a, b)$ on the unit circle, the angle $ heta$ can be computed directly using trigonometric functions.
Hence, $ heta = ext{atan2}(b, a)$.
Answer: $ heta = ext{atan2}(b, a)$.
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