Calculate the sine, cosine and tangent values for angles on the unit circle

Let’s calculate the sine, cosine, and tangent values of the angle $\frac{5\pi}{6}$ on the unit circle.

First, determine the coordinates of the angle $\frac{5\pi}{6}$.

Since $\frac{5\pi}{6} = 180^\circ – 30^\circ$, it is in the second quadrant where sine is positive, and cosine is negative.

Coordinates of $30^\circ$ are $(\cos 30^\circ, \sin 30^\circ) = (\frac{\sqrt{3}}{2}, \frac{1}{2})$.

Therefore, the coordinates of $\frac{5\pi}{6}$ are $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.

Thus,

$\sin \left(\frac{5\pi}{6}\right) = \frac{1}{2}$

$\cos \left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$

For tangent,

$\tan \left(\frac{5\pi}{6}\right) = \frac{\sin \left(\frac{5\pi}{6}\right)}{\cos \left(\frac{5\pi}{6}\right)} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

Consider the angle $frac{7pi}{4}$ on the unit circle. We need to find the sine, cosine, and tangent values.

First, identify the quadrant:

$frac{7pi}{4} = 2pi – frac{pi}{4}$, which puts it in the fourth quadrant where sine is negative, and cosine is positive.

Coordinates of $frac{pi}{4}$ are $(cos frac{pi}{4}, sin frac{pi}{4}) = left(frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight)$.

Thus, the coordinates of $frac{7pi}{4}$ are $left(frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}
ight)$.

Thus,

$sin left(frac{7pi}{4}
ight) = -frac{sqrt{2}}{2}$

$cos left(frac{7pi}{4}
ight) = frac{sqrt{2}}{2}$

For tangent,

$ an left(frac{7pi}{4}
ight) = frac{sin left(frac{7pi}{4}
ight)}{cos left(frac{7pi}{4}
ight)} = frac{-frac{sqrt{2}}{2}}{frac{sqrt{2}}{2}} = -1$

Calculate the sine, cosine, and tangent values for $frac{3pi}{2}$.

At $frac{3pi}{2}$ (270 degrees) on the unit circle,

$sin left(frac{3pi}{2}
ight) = -1$

$cos left(frac{3pi}{2}
ight) = 0$

Tangent is undefined as cosine is zero:

$ an left(frac{3pi}{2}
ight) = ext{undefined}$

Start Using PopAi Today