Find the exact values of $sin( heta)$ and $cos( heta)$ for $ heta = frac{5pi}{6}$ using the unit circle.
Answer 1
Emily Hall
To find the exact values of $\sin(\theta)$ and $\cos(\theta)$ for $\theta = \frac{5\pi}{6}$, we use the unit circle.
The angle $\frac{5\pi}{6}$ radians is in the second quadrant, where sine is positive and cosine is negative.
The reference angle for $\frac{5\pi}{6}$ is $\pi – \frac{5\pi}{6} = \frac{\pi}{6}$.
From the unit circle, we know the coordinates for $\frac{\pi}{6}$ are $\left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right)$.
Therefore, for $\frac{5\pi}{6}$, the coordinates are $\left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right)$.
Hence, $\sin(\frac{5\pi}{6}) = \frac{1}{2}$ and $\cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2}$.
Answer 2
Thomas Walker
Let’s determine $sin( heta)$ and $cos( heta)$ for $ heta = frac{5pi}{6}$ using the unit circle.
Since $frac{5pi}{6}$ is in the second quadrant, sine is positive and cosine is negative.
Consider the reference angle $pi – frac{5pi}{6} = frac{pi}{6}$.
The coordinates for $frac{pi}{6}$ on the unit circle are $left( frac{sqrt{3}}{2}, frac{1}{2}
ight)$.
Thus, for $frac{5pi}{6}$, the coordinates are $left( -frac{sqrt{3}}{2}, frac{1}{2}
ight)$.
Therefore, $sin(frac{5pi}{6}) = frac{1}{2}$ and $cos(frac{5pi}{6}) = -frac{sqrt{3}}{2}$.
Answer 3
William King
For $ heta = frac{5pi}{6}$,
$sin( heta) = frac{1}{2}$
and $cos( heta) = -frac{sqrt{3}}{2}$ from the unit circle.
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