Use reference angles to find coordinates on the unit circle

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Use reference angles to find coordinates on the unit circle. Question: Find the coordinates of the point on the unit circle at an angle of θ. Give your answer in the form (x, y) and leave any fractions in fraction form. Provide your answer below: FEEDBACK MORE INSTRUCTIONS Content attribution

The Correct Answer and Explanation is:

To find the coordinates of the point on the unit circle at a given angle θ\thetaθ, we use the fact that for any point on the unit circle, the coordinates (x,y)(x, y)(x,y) correspond to (cos⁡θ,sin⁡θ)(\cos \theta, \sin \theta)(cosθ,sinθ). The unit circle has a radius of 1, and for any angle θ\thetaθ, these trigonometric functions give the xxx- and yyy-coordinates of the point on the circle.

Steps to Find the Coordinates Using Reference Angles:

  1. Identify the Reference Angle:
    A reference angle is the acute angle formed by the terminal side of the angle θ\thetaθ and the x-axis. The reference angle helps us determine the cosine and sine values for angles in different quadrants.
  2. Determine the Quadrant:
    The quadrant in which the angle lies will affect the signs of cos⁡θ\cos \thetacosθ and sin⁡θ\sin \thetasinθ:
    • Quadrant I: Both cos⁡θ\cos \thetacosθ and sin⁡θ\sin \thetasinθ are positive.
    • Quadrant II: cos⁡θ\cos \thetacosθ is negative, and sin⁡θ\sin \thetasinθ is positive.
    • Quadrant III: Both cos⁡θ\cos \thetacosθ and sin⁡θ\sin \thetasinθ are negative.
    • Quadrant IV: cos⁡θ\cos \thetacosθ is positive, and sin⁡θ\sin \thetasinθ is negative.
  3. Find the Cosine and Sine:
    Using the reference angle, find the cosine and sine values. These values will then be adjusted based on the quadrant in which θ\thetaθ lies.
  4. Write the Coordinates:
    Once you have the cosine and sine values for the angle θ\thetaθ, write the coordinates of the point on the unit circle as (x,y)=(cos⁡θ,sin⁡θ)(x, y) = (\cos \theta, \sin \theta)(x,y)=(cosθ,sinθ).

Example:

Suppose θ=120∘\theta = 120^\circθ=120∘:

  • The reference angle is 180∘−120∘=60∘180^\circ – 120^\circ = 60^\circ180∘−120∘=60∘.
  • The cosine and sine values for 60∘60^\circ60∘ are:
    • cos⁡60∘=12\cos 60^\circ = \frac{1}{2}cos60∘=21​
    • sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2}sin60∘=23​​
  • Since 120∘120^\circ120∘ is in the second quadrant, cos⁡θ\cos \thetacosθ will be negative and sin⁡θ\sin \thetasinθ will be positive.
  • Thus, the coordinates are (−12,32)\left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)(−21​,23​​).

The final coordinates of the point on the unit circle for θ=120∘\theta = 120^\circθ=120∘ are: (−12,32)\left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)(−21​,23​​)

This is how you can use reference angles to find coordinates on the unit circle. Let me know if you have a specific angle in mind for a similar calculation!

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