Find the values of $ an( heta) $ at various angles and verify using the unit circle

To find the values of $ \tan(\theta) $ at various angles and verify using the unit circle, we consider the following angles: $ \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} $

1. For $ \theta = \frac{\pi}{4} $:

$ \tan(\frac{\pi}{4}) = 1 $

2. For $ \theta = \frac{3\pi}{4} $:

$ \tan(\frac{3\pi}{4}) = -1 $

3. For $ \theta = \frac{5\pi}{4} $:

$ \tan(\frac{5\pi}{4}) = 1 $

4. For $ \theta = \frac{7\pi}{4} $:

$ \tan(\frac{7\pi}{4}) = -1 $

Verification: Using the unit circle, we observe that at these angles, the tangent value is consistent with the coordinates (x, y) where $ \tan(\theta) = \frac{y}{x} $.

To find the values of $ an( heta) $ at various angles and verify using the unit circle, we consider the following angles: $ heta = frac{pi}{3}, frac{2pi}{3}, frac{4pi}{3}, frac{5pi}{3} $

1. For $ heta = frac{pi}{3} $:

$ an(frac{pi}{3}) = sqrt{3} $

2. For $ heta = frac{2pi}{3} $:

$ an(frac{2pi}{3}) = -sqrt{3} $

3. For $ heta = frac{4pi}{3} $:

$ an(frac{4pi}{3}) = sqrt{3} $

4. For $ heta = frac{5pi}{3} $:

$ an(frac{5pi}{3}) = -sqrt{3} $

Verification: Using the unit circle, we can see that at these angles, the tangent value corresponds to the coordinate ratio $ frac{y}{x} $.

To find the values of $ an( heta) $ at various angles and verify using the unit circle, we consider the following angles: $ heta = 0, pi, frac{pi}{2}, frac{3pi}{2} $

1. For $ heta = 0 $:

$ an(0) = 0 $

2. For $ heta = pi $:

$ an(pi) = 0 $

3. For $ heta = frac{pi}{2} $:

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

4. For $ heta = frac{3pi}{2} $:

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

Verification: Using the unit circle at these

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