Find the value of the cosecant function for an angle in the unit circle.

Answer 1:

Given an angle \( \theta \) in the unit circle, we need to find the value of \( \csc(\theta) \). Recall that \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

Let’s consider \( \theta = \frac{5\pi}{6} \). First, we find \( \sin\left(\frac{5\pi}{6}\right) \). Since \( \sin\left(\frac{5\pi}{6}\right) = \sin\left(\pi – \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) \), we have \( \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \).

Thus, \( \csc\left(\frac{5\pi}{6}\right) = \frac{1}{\sin\left(\frac{5\pi}{6}\right)} = \frac{1}{\frac{1}{2}} = 2 \).

Answer 2:

Consider ( heta = frac{7pi}{4} ) on the unit circle. To find ( cscleft(frac{7pi}{4}
ight) ), we must first determine ( sinleft(frac{7pi}{4}
ight) ). Since ( frac{7pi}{4} = 2pi – frac{pi}{4} ), and ( sin(2pi – x) = -sin(x) ), we have ( sinleft(frac{7pi}{4}
ight) = -sinleft(frac{pi}{4}
ight) = -frac{sqrt{2}}{2} ).

Therefore, ( cscleft(frac{7pi}{4}
ight) = frac{1}{-sinleft(frac{7pi}{4}
ight)} = frac{1}{-frac{sqrt{2}}{2}} = -sqrt{2} ).

Answer 3:

Given ( heta = frac{3pi}{2} ), we need ( cscleft(frac{3pi}{2}
ight) ). Since ( sinleft(frac{3pi}{2}
ight) = -1 ), it follows that ( cscleft(frac{3pi}{2}
ight) = frac{1}{-1} = -1 ).

Start Using PopAi Today