The ASTC Rule
“All Students Take Calculus” — a mnemonic that tells you which trig functions are positive in each quadrant. Pair this with reference angles and you can evaluate any trig function at any angle.
Introduction
In Quadrant I, all trig functions are positive. But what about the other three quadrants? The ASTC rule answers this question by telling you exactly which functions remain positive in each region. Everything else is negative.
Past Knowledge
On the unit circle, and . Coordinate signs depend on the quadrant.
Today's Goal
Determine the sign of any trig function in any quadrant using the ASTC mnemonic.
Future Success
ASTC + reference angles = the complete evaluation strategy for any trig function at any angle.
Key Concepts
The ASTC Mnemonic
“All Students Take Calculus” tells you which trig functions are positive in each quadrant:
QII — S
Sine positive
(and cosecant)
QI — A
All positive
QIII — T
Tangent positive
(and cotangent)
QIV — C
Cosine positive
(and secant)
Why Does This Work?
This follows directly from the signs of the coordinates in each quadrant:
| Quadrant | x (cos) | y (sin) | y/x (tan) |
|---|---|---|---|
| I | + | + | + |
| II | − | + | − |
| III | − | − | + |
| IV | + | − | − |
, so tangent is positive when and have the same sign (both positive in QI, both negative in QIII).
Worked Examples
Determining the Sign
Question: Is positive or negative?
Step 1: is in Quadrant III.
Step 2: ASTC says only Tangent is positive in QIII.
Step 3: Cosine is NOT tangent, so it is negative.
Final Answer: is negative.
Attaching a Sign to a Value
Question: Using the reference angle, evaluate .
Step 1: Find the reference angle. is in QII. .
Step 2: Find the Quadrant I value. .
Step 3: Apply the ASTC sign. In QII, Sine is positive.
Final Answer:
Full Evaluation with ASTC
Question: Evaluate .
Step 1: Find the quadrant. is between and → Quadrant IV.
Step 2: Find the reference angle.
Step 3: Q1 value.
Step 4: ASTC sign. In QIV, Cosine is positive.
Final Answer:
Common Pitfalls
Forgetting that Reciprocal Functions Share the Same Sign
Sine and cosecant always have the same sign. Cosine and secant always have the same sign. Tangent and cotangent always have the same sign. This is because — a positive number divided into 1 stays positive, and a negative stays negative.
Real-Life Applications
Physics: Projectile Motion Components
When a ball is thrown at an angle, its velocity has horizontal () and vertical () components. A projectile traveling into Quadrant II (upward and to the left) has negative horizontal velocity (cosine is negative in QII) but positive vertical velocity (sine is positive in QII). The ASTC rule is precisely how physics engines decompose vectors.
Practice Quiz
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