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Difficulty: MediumUnit Circle and Angle Measures (Radians and Degrees)

For an angle θ\theta in standard position, match each description of its terminal side on the left with the corresponding coordinates of its intersection point on the unit circle on the right.

  • The terminal side of θ\theta lies in Quadrant II with a reference angle of 60∘60^\circ.(−12,32)(-\frac{1}{2}, \frac{\sqrt{3}}{2})
  • The terminal side of θ\theta lies in Quadrant III with a reference angle of 45∘45^\circ.(−22,−22)(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})
  • The terminal side of θ\theta lies in Quadrant IV with a reference angle of 30∘30^\circ.(32,−12)(\frac{\sqrt{3}}{2}, -\frac{1}{2})
  • The terminal side of θ\theta lies in Quadrant III with a reference angle of 30∘30^\circ.(−32,−12)(-\frac{\sqrt{3}}{2}, -\frac{1}{2})

Answer

The terminal side in Quadrant II with reference angle 60∘60^\circ matches (−12,32)(-\frac{1}{2}, \frac{\sqrt{3}}{2}); in Quadrant III with reference angle 45∘45^\circ matches (−22,−22)(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}); in Quadrant IV with reference angle 30∘30^\circ matches (32,−12)(\frac{\sqrt{3}}{2}, -\frac{1}{2}); in Quadrant III with reference angle 30∘30^\circ matches (−32,−12)(-\frac{\sqrt{3}}{2}, -\frac{1}{2}).
Each terminal side is matched correctly to its coordinates by applying the quadrant signs to the trigonometric values of the reference angles. Quadrant II corresponds to (−,+)(-, +), Quadrant III corresponds to (−,−)(-, -), and Quadrant IV corresponds to (+,−)(+, -). Using standard unit circle coordinates, a 60∘60^\circ reference angle gives magnitudes of (12,32)(\frac{1}{2}, \frac{\sqrt{3}}{2}), a 45∘45^\circ reference angle gives (22,22)(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}), and a 30∘30^\circ reference angle gives (32,12)(\frac{\sqrt{3}}{2}, \frac{1}{2}).

Step-by-Step Solution

1
Determine the signs of the xx- and yy-coordinates based on the quadrant of the terminal side.
Quadrant II points have (−,+)(-, +) coordinates; Quadrant III points have (−,−)(-, -) coordinates; Quadrant IV points have (+,−)(+, -) coordinates.
On the unit circle, x=cos⁡θx = \cos\theta and y=sin⁡θy = \sin\theta. Cosine is negative in Quadrants II and III, while sine is negative in Quadrants III and IV.
2
Find the absolute values of the coordinates using the reference angle.
A 30∘30^\circ reference angle corresponds to coordinates of magnitude (32,12)(\frac{\sqrt{3}}{2}, \frac{1}{2}); a 45∘45^\circ reference angle corresponds to (22,22)(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}); a 60∘60^\circ reference angle corresponds to (12,32)(\frac{1}{2}, \frac{\sqrt{3}}{2}).
The reference angle determines the basic trigonometric values cos⁡θref\cos\theta_{\text{ref}} and sin⁡θref\sin\theta_{\text{ref}}.
3
Combine the quadrant signs and coordinate magnitudes to find the unique point.
Quadrant II with 60∘60^\circ reference angle is (−12,32)(-\frac{1}{2}, \frac{\sqrt{3}}{2}). Quadrant III with 45∘45^\circ reference angle is (−22,−22)(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}). Quadrant IV with 30∘30^\circ reference angle is (32,−12)(\frac{\sqrt{3}}{2}, -\frac{1}{2}). Quadrant III with 30∘30^\circ reference angle is (−32,−12)(-\frac{\sqrt{3}}{2}, -\frac{1}{2}).
Applying the quadrant signs from Step 1 to the magnitude values from Step 2 yields the exact coordinates on the unit circle.

Key Concept

Coordinates of points on the unit circle are given by (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta), where the magnitude is determined by the reference angle and the signs are determined by the quadrant of the angle.
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Topics

Unit Circle and Angle Measures (Radians and Degrees)

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