Step by Step
A
Quadrant I — All positive
In Quadrant I (both x and y positive), all six trig functions are positive.
S
Quadrant II — Sine positive
In Quadrant II, only sine (and its reciprocal, cosecant) are positive; cosine and tangent are negative here.
T
Quadrant III — Tangent positive
In Quadrant III, only tangent (and its reciprocal, cotangent) are positive; sine and cosine are negative here.
C
Quadrant IV — Cosine positive
In Quadrant IV, only cosine (and its reciprocal, secant) are positive; sine and tangent are negative here.
Applied Walkthrough
1
Determine the sign of sin(200°). First, figure out which quadrant 200° falls in: since 180° starts Quadrant III and 270° starts Quadrant IV, 200° falls in Quadrant III.
2
Using ASTC ("All Students Take Calculus"), Quadrant III means only Tangent (and cotangent) are positive — sine and cosine are both negative there.
3
So sin(200°) is negative.
4
As a check: sin(200°) = sin(180°+20°), and since 200° is past the x-axis in the lower half of the circle, the y-coordinate (which sine represents) is indeed negative — confirming the ASTC prediction.
Exam Application
Exams test whether you can quickly determine which quadrant an angle falls in and then correctly recall — using ASTC — whether a given trig function is positive or negative in that quadrant.
⚠ Common Trap
The most common trap is memorizing ASTC but forgetting it also applies to the reciprocal functions — cosecant follows sine's sign, secant follows cosine's sign, and cotangent follows tangent's sign.
✓ Quick Self-Check
1. What does the ASTC mnemonic ("All Students Take Calculus") represent?
Which trig functions are positive in each quadrant: All (Q1), Sine (Q2), Tangent (Q3), Cosine (Q4).
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2. Which trig functions are positive in Quadrant II?
Sine and cosecant.
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3. What is the sign of sin(200°)?
Negative, since 200° falls in Quadrant III, where only tangent is positive.
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4. Which quadrant has only tangent and cotangent positive?
Quadrant III.
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5. In Quadrant IV, which functions are positive?
Cosine and secant.
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