Find the values of $cot( heta)$ for a given angle on the unit circle and verify their consistency
Answer 1
Benjamin Clark
To find the values of $\cot(\theta)$ for $\theta = \frac{3\pi}{4}$ on the unit circle, we start by identifying the coordinates of this angle on the unit circle.
The coordinates for $\theta = \frac{3\pi}{4}$ are $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$. The cotangent function is defined as the cosine divided by the sine of the angle: $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$.
Therefore,
$\cot\left(\frac{3\pi}{4}\right) = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1$
The value of $\cot\left(\frac{3\pi}{4}\right)$ is -1.
Answer 2
Daniel Carter
Let’s consider the angle $ heta = frac{5pi}{6}$ on the unit circle. We will determine $cot( heta)$ and confirm its correctness.
The coordinates for $ heta = frac{5pi}{6}$ are $left(-frac{sqrt{3}}{2}, frac{1}{2}
ight)$. The cotangent function $cot( heta)$ is given by $cot( heta) = frac{cos( heta)}{sin( heta)}$.
Hence,
$cotleft(frac{5pi}{6}
ight) = frac{-frac{sqrt{3}}{2}}{frac{1}{2}} = -sqrt{3}$
The value of $cotleft(frac{5pi}{6}
ight)$ is -sqrt{3}.
Answer 3
Ava Martin
Considering the angle $ heta = frac{7pi}{6}$ on the unit circle, we directly compute the value of $cot( heta)$.
The coordinates for $ heta = frac{7pi}{6}$ are $left(-frac{sqrt{3}}{2}, -frac{1}{2}
ight)$. Therefore,
$cotleft(frac{7pi}{6}
ight) = frac{-frac{sqrt{3}}{2}}{-frac{1}{2}} = sqrt{3}$
The value of $cotleft(frac{7pi}{6}
ight)$ is sqrt{3}.
Start Using PopAi Today