Calculate the trigonometric values at specific angles on the unit circle
Answer 1
Emily Hall
Given the angle $ \theta = \frac{3\pi}{4} $, find the values of $ \sin \theta $, $ \cos \theta $, and $ \tan \theta $.
Step 1: Identify the reference angle. The reference angle for $ \theta = \frac{3\pi}{4} $ is $ \frac{\pi}{4} $.
Step 2: Determine the sine, cosine, and tangent values of the reference angle. For $ \theta = \frac{\pi}{4} $, $ \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} $, $ \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} $, and $ \tan \frac{\pi}{4} = 1 $.
Step 3: Determine the signs of these values in the second quadrant. In the second quadrant, sine is positive, cosine is negative, and tangent is negative.
Therefore, $ \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} $, $ \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} $, and $ \tan \frac{3\pi}{4} = -1 $.
Answer 2
Ella Lewis
Given the angle $ heta = frac{5pi}{6} $, find the values of $ sin heta $, $ cos heta $, and $ an heta $.
Step 1: Identify the reference angle. The reference angle for $ heta = frac{5pi}{6} $ is $ frac{pi}{6} $.
Step 2: Determine the sine, cosine, and tangent values of the reference angle. For $ heta = frac{pi}{6} $, $ sin frac{pi}{6} = frac{1}{2} $, $ cos frac{pi}{6} = frac{sqrt{3}}{2} $, and $ an frac{pi}{6} = frac{1}{sqrt{3}} $.
Step 3: Determine the signs of these values in the second quadrant. In the second quadrant, sine is positive, cosine is negative, and tangent is negative.
Therefore, $ sin frac{5pi}{6} = frac{1}{2} $, $ cos frac{5pi}{6} = -frac{sqrt{3}}{2} $, and $ an frac{5pi}{6} = -frac{1}{sqrt{3}} $.
Answer 3
Lucas Brown
Given the angle $ heta = frac{7pi}{4} $, find the values of $ sin heta $, $ cos heta $, and $ an heta $.
Step 1: Identify the reference angle. The reference angle for $ heta = frac{7pi}{4} $ is $ frac{pi}{4} $.
Step 2: Determine the sine, cosine, and tangent values of the reference angle. For $ heta = frac{pi}{4} $, $ sin frac{pi}{4} = frac{sqrt{2}}{2} $, $ cos frac{pi}{4} = frac{sqrt{2}}{2} $, and $ an frac{pi}{4} = 1 $.
Step 3: Determine the signs of these values in the fourth quadrant. In the fourth quadrant, sine is negative, cosine is positive, and tangent is negative.
Therefore, $ sin frac{7pi}{4} = -frac{sqrt{2}}{2} $, $ cos frac{7pi}{4} = frac{sqrt{2}}{2} $, and $ an frac{7pi}{4} = -1 $.
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