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ASTC Rule — Definition, Formula & Examples

The ASTC Rule is a mnemonic — "All Students Take Calculus" — that tells you which trigonometric functions are positive in each of the four quadrants of the coordinate plane.

ASTC assigns a letter to each quadrant in counter-clockwise order starting from Quadrant I: A (All) means sine, cosine, and tangent are all positive; S (Sine) means only sine is positive; T (Tangent) means only tangent is positive; C (Cosine) means only cosine is positive. The reciprocal functions (csc, sec, cot) share the sign of their corresponding primary function.

How It Works

Label the four quadrants starting top-right and moving counter-clockwise: I → A, II → S, III → T, IV → C. In Quadrant I (A for All), every trig ratio is positive. In Quadrant II (S for Sine), only sine (and its reciprocal, cosecant) are positive. In Quadrant III (T for Tangent), only tangent (and cotangent) are positive. In Quadrant IV (C for Cosine), only cosine (and secant) are positive. Any function not listed as positive in a given quadrant is negative there.

Worked Example

Problem: Determine the sign of sin 210° and cos 210°.
Identify the quadrant: 210° lies between 180° and 270°, so the angle is in Quadrant III.
180°<210°<270°  ⟹  Quadrant III180° < 210° < 270° \implies \text{Quadrant III}
Apply the ASTC Rule: Quadrant III corresponds to T (Tangent). Only tangent is positive here, so sine and cosine are both negative.
Confirm with reference angle: The reference angle is 30°. Using ASTC, attach a negative sign to both sine and cosine.
sin⁡210°=−sin⁡30°=−12,cos⁡210°=−cos⁡30°=−32\sin 210° = -\sin 30° = -\tfrac{1}{2}, \quad \cos 210° = -\cos 30° = -\tfrac{\sqrt{3}}{2}
Answer: sin⁡210°=−12\sin 210° = -\tfrac{1}{2} (negative) and cos⁡210°=−32\cos 210° = -\tfrac{\sqrt{3}}{2} (negative), consistent with only tangent being positive in Quadrant III.

Why It Matters

When you evaluate trig functions beyond 90° or work with inverse trig, you need the correct sign quickly. The ASTC Rule lets you attach the right sign without re-deriving it from the unit circle each time, which saves effort on timed exams in precalculus and calculus.

Common Mistakes

Mistake: Forgetting that "T" in Quadrant III means only tangent is positive, and assuming sine is also positive because sine appears in the tangent ratio.
Correction: In Quadrant III both sine and cosine are negative. Their ratio (tangent) is positive precisely because a negative divided by a negative yields a positive.

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