Find the value of $csc( heta)$ for a given angle on the unit circle
Answer 1
Christopher Garcia
Given that the angle \(\theta\) is \(\frac{5\pi}{6}\), find the value of \(csc(\theta)\) using the unit circle.
Step 1: First, locate the angle \(\frac{5\pi}{6}\) on the unit circle. This angle is in the second quadrant.
Step 2: The reference angle for \(\frac{5\pi}{6}\) is \(\frac{\pi}{6}\).
Step 3: The sine of \(\frac{5\pi}{6}\) is equal to the sine of \(\frac{\pi}{6}\) because they share the same reference angle.
Step 4: Thus, \(\sin(\frac{5\pi}{6}) = \sin(\frac{\pi}{6}) = \frac{1}{2}\).
Step 5: The cosecant function is the reciprocal of the sine function. Therefore, \(\csc(\frac{5\pi}{6}) = \frac{1}{\sin(\frac{5\pi}{6})} = \frac{1}{\frac{1}{2}} = 2\).
Answer 2
Matthew Carter
Given ( heta = frac{7pi}{4}), determine the value of (csc( heta)) using the unit circle.
Step 1: Locate the angle (frac{7pi}{4}) on the unit circle. This angle is in the fourth quadrant.
Step 2: The reference angle for (frac{7pi}{4}) is (frac{pi}{4}).
Step 3: The sine of (frac{7pi}{4}) is the negative sine of (frac{pi}{4}) because it is in the fourth quadrant.
Step 4: Thus, (sin(frac{7pi}{4}) = -sin(frac{pi}{4}) = -frac{sqrt{2}}{2}).
Step 5: Therefore, (csc(frac{7pi}{4}) = frac{1}{sin(frac{7pi}{4})} = frac{1}{-frac{sqrt{2}}{2}} = -frac{2}{sqrt{2}} = -sqrt{2}).
Answer 3
Alex Thompson
Given ( heta = frac{2pi}{3}), find (csc( heta)).
(frac{2pi}{3}) is in the second quadrant.
The reference angle is (frac{pi}{3}).
(sin(frac{2pi}{3}) = sin(frac{pi}{3}) = frac{sqrt{3}}{2}).
(csc(frac{2pi}{3}) = frac{1}{sin(frac{2pi}{3})} = frac{1}{frac{sqrt{3}}{2}} = frac{2}{sqrt{3}} = frac{2sqrt{3}}{3}).
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