Read Sine and Cosine Signs from Coordinates

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Angle 30° terminates strictly inside quadrant I. What are the signs of sine and cosine? Candidate answers: (A) sin negative, cos negative (B) sin negative, cos positive (C) sin positive, cos negative (D) sin positive, cos positive.
• Independent check inputs: Angle 140° terminates strictly inside quadrant II. What are the signs of sine and cosine? Candidate answers: (A) sin negative, cos negative (B) sin negative, cos positive (C) sin positive, cos negative (D) sin positive, cos positive.

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. Quadrant location fixes the signs of a point’s horizontal and vertical coordinates. Since cosine is horizontal and sine vertical on the unit circle, those coordinate signs determine the trigonometric signs.

Here is the method on the supplied teaching case. After removing whole turns if needed, the angle remains in quadrant I. Sine is its y-coordinate (positive) and cosine its x-coordinate (positive); it is not on an axis.

Angles on coordinate axes require separate zero-value cases, so this exercise excludes them. Keep sine tied to y and cosine tied to x; positive angle direction alone does not determine both signs.

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: Angle 700° terminates strictly inside quadrant IV. What are the signs of sine and cosine? Candidate answers: (A) sin negative, cos negative (B) sin negative, cos positive (C) sin positive, cos negative (D) sin positive, cos positive. Show the mathematical reason, not only the selected answer. Expected result: sin negative, cos positive. After removing whole turns if needed, the angle remains in quadrant IV. Sine is its y-coordinate (negative) and cosine its x-coordinate (positive); it is not on an axis.

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