Find Exact Coordinates from a Special Reference Angle

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Using special-angle triangles, find the unit-circle point at angle π/6. Candidate answers: (A) (1/2, √3/2) (B) (−1/2, √3/2) (C) (−√3/2, 1/2) (D) (√3/2, 1/2).
• Independent check inputs: Using special-angle triangles, find the unit-circle point at angle π/3. Candidate answers: (A) (1/2, √3/2) (B) (−1/2, √3/2) (C) (−√3/2, 1/2) (D) (√3/2, 1/2).

Learn the idea

Radians measure an angle by arc length relative to radius. On the unit circle, cosine and sine are horizontal and vertical coordinates, so quadrant signs and special triangles support exact values beyond acute angles. The unit-circle point at angle θ is (cosθ,sinθ). A thirty-sixty-ninety triangle supplies coordinate magnitudes one half and square root of three over two, with quadrant signs locating the point.

Here is the method on the supplied teaching case. The point is (cos θ,sin θ). Use the 30°–60°–90° coordinate magnitudes 1/2 and √3/2, then apply the quadrant signs; the ordered pair is (√3/2, 1/2).

Cosine comes first in the coordinate pair and sine second. Determine the reference angle and quadrant separately; swapping magnitudes or forgetting a sign can produce a different legitimate point.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Using special-angle triangles, find the unit-circle point at angle 11π/6. Candidate answers: (A) (1/2, √3/2) (B) (−1/2, √3/2) (C) (√3/2, 1/2) (D) (√3/2, −1/2). Show the mathematical reason, not only the selected answer. Expected result: (√3/2, −1/2). The point is (cos θ,sin θ). Use the 30°–60°–90° coordinate magnitudes 1/2 and √3/2, then apply the quadrant signs; the ordered pair is (√3/2, −1/2).

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