Finding Specific Trigonometric Values on the Unit Circle
Answer 1
Henry Green
Given the angle θ = 225°, find the sine, cosine, and tangent values using the unit circle.
First, convert the angle 225° to radians: $225° = \frac{5\pi}{4}$ radians.
On the unit circle, the angle $\frac{5\pi}{4}$ is located in the third quadrant, where sine and cosine are both negative.
The coordinates for $\frac{5\pi}{4}$ are $(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$.
Thus, $\sin(225°) = -\frac{\sqrt{2}}{2}$, $\cos(225°) = -\frac{\sqrt{2}}{2}$
Finally, the tangent value: $\tan(225°) = \frac{\sin(225°)}{\cos(225°)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1$.
Therefore, $\sin(225°) = -\frac{\sqrt{2}}{2}$, $\cos(225°) = -\frac{\sqrt{2}}{2}$, $\tan(225°) = 1$.
Answer 2
John Anderson
Given the angle θ = 300°, calculate the sine, cosine, and tangent values using the unit circle.
First, convert the angle 300° to radians: $300° = frac{5pi}{3}$ radians.
On the unit circle, the angle $frac{5pi}{3}$ is located in the fourth quadrant, where sine is negative and cosine is positive.
The coordinates for $frac{5pi}{3}$ are $(frac{1}{2}, -frac{sqrt{3}}{2})$.
Thus, $sin(300°) = -frac{sqrt{3}}{2}$, $cos(300°) = frac{1}{2}$
Finally, the tangent value: $ an(300°) = frac{sin(300°)}{cos(300°)} = frac{-frac{sqrt{3}}{2}}{frac{1}{2}} = -sqrt{3}$.
Therefore, $sin(300°) = -frac{sqrt{3}}{2}$, $cos(300°) = frac{1}{2}$, $ an(300°) = -sqrt{3}$.
Answer 3
James Taylor
Given the angle θ = 120°, find the sine, cosine, and tangent values using the unit circle.
First, convert the angle 120° to radians: $120° = frac{2pi}{3}$ radians.
The coordinates for $frac{2pi}{3}$ are $(-frac{1}{2}, frac{sqrt{3}}{2})$.
Thus, $sin(120°) = frac{sqrt{3}}{2}$, $cos(120°) = -frac{1}{2}$
Finally, the tangent value: $ an(120°) = frac{sin(120°)}{cos(120°)} = frac{frac{sqrt{3}}{2}}{-frac{1}{2}} = -sqrt{3}$.
Therefore, $sin(120°) = frac{sqrt{3}}{2}$, $cos(120°) = -frac{1}{2}$, $ an(120°) = -sqrt{3}$.
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