Find the slope of the tangent line to the unit circle at the point where $ heta = frac{pi}{4}$.
Answer 1
Abigail Nelson
To find the slope of the tangent line to the unit circle at the point where $\theta = \frac{\pi}{4}$, we start by finding the coordinates of the point on the unit circle.
At $\theta = \frac{\pi}{4}$, the coordinates are:
$ (\cos(\frac{\pi}{4}), \sin(\frac{\pi}{4})) = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
The slope of the tangent line to the unit circle at any point $(x, y)$ is given by $-\frac{x}{y}$.
Therefore, the slope at the point $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ is:
$ -\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1 $
Answer 2
Daniel Carter
To determine the slope of the tangent line at $ heta = frac{pi}{4}$ on the unit circle, we need the coordinates corresponding to this angle. These coordinates are:
$ (cos(frac{pi}{4}), sin(frac{pi}{4})) = (frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$
The slope formula for a tangent to the unit circle at $(x, y)$ is $-frac{x}{y}$.
So, substituting $x = frac{sqrt{2}}{2}$ and $y = frac{sqrt{2}}{2}$:
$ -frac{frac{sqrt{2}}{2}}{frac{sqrt{2}}{2}} = -1 $
Answer 3
Lily Perez
To find the slope of the tangent line at $ heta = frac{pi}{4}$, first find the point on the unit circle:
$ (frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$
Then use the slope formula $-frac{x}{y}$:
$ -1 $
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