Find the value of $ an(θ) $ using the unit circle when $ θ $ is in the third quadrant
Answer 1
Alex Thompson
To find the value of $ \tan(θ) $ using the unit circle, we need to determine the coordinates where $ θ $ intersects the unit circle in the third quadrant.
In the third quadrant, both the x and y coordinates are negative. Suppose $ θ = 225° $ (or $ \frac{5π}{4} $ in radians). In this case, the coordinates on the unit circle are $ ( -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} ) $.
The tangent of $ θ $ is given by the ratio of the y-coordinate to the x-coordinate:
$ \tan(225°) = \frac{y}{x} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 $
Answer 2
Lily Perez
To determine $ an(θ) $ using the unit circle in the third quadrant, we identify the coordinates for an angle in this quadrant. For $ θ = 240° $ (or $ frac{4π}{3} $ radians), the corresponding coordinates are $ ( -frac{1}{2}, -frac{sqrt{3}}{2} ) $.
The tangent value is calculated as:
$ an(240°) = frac{y}{x} = frac{-frac{sqrt{3}}{2}}{-frac{1}{2}} = sqrt{3} $
Answer 3
Thomas Walker
To find $ an(θ) $ in the third quadrant using the unit circle, take $ θ = 210° $ (or $ frac{7π}{6} $ radians). The coordinates are $ ( -frac{sqrt{3}}{2}, -frac{1}{2} ) $.
Thus,
$ an(210°) = frac{y}{x} = frac{-frac{1}{2}}{-frac{sqrt{3}}{2}} = frac{1}{sqrt{3}} $
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