The Quadrants and Signs of Trigonometric Ratios
Simple Explanation
Since x and y can each be positive or negative depending on the quadrant, so can sin θ, cos θ, and tan θ. A quick rule (often remembered as "All Students Take Calculus," for quadrants I-IV) says: Quadrant I — all three positive; II — only sine positive; III — only tangent positive; IV — only cosine positive.
Why Do We Need It?
Knowing the correct sign instantly, without recalculating from scratch, prevents a very common source of errors when evaluating trigonometric ratios for angles outside Quadrant I.
See It
A coordinate plane divided into four quadrants, each labelled with which trigonometric ratio is positive there: All, Sin, Tan, Cos
Formula
The Quadrant Sign Rule (ASTC)
Quadrant I: All positive. II: Sin (and cosec) positive. III: Tan (and cot) positive. IV: Cos (and sec) positive.
A quick way to remember which trigonometric ratio is positive in each quadrant — often remembered with a phrase like "All Students Take Calculus," reading the quadrants in order I, II, III, IV.
- Quadrant I
- — 0° to 90° — x and y both positive
- Quadrant II
- — 90° to 180° — x negative, y positive
- Quadrant III
- — 180° to 270° — x and y both negative
- Quadrant IV
- — 270° to 360° — x positive, y negative
When to use it: Whenever you need to determine whether a trigonometric ratio should be positive or negative for a given angle.
Worked Example
Identify a quadrant from two ratio signs
In which quadrant is cos θ negative but sin θ positive?
Why Does This Work?
The sign of each ratio comes directly from the signs of x and y in that quadrant (sin θ=y/r and r is always positive, so sin θ's sign matches y's; similarly cos θ's sign matches x's; and tan θ=y/x is positive exactly when x and y share the same sign).
Real-Life Example
Reading the phase of an oscillating signal
An engineer analyzing an alternating current signal needs to know quickly whether a component is currently positive or negative, based only on its phase angle.
The ASTC rule gives the sign instantly from the phase angle's quadrant, without recomputing the full trigonometric value.
Practice
If tan θ is positive and cos θ is negative, in which quadrant does θ lie?
MediumCommon mistake
Assuming tan follows some separate, independent rule — tan θ = sin θ/cos θ, so its sign is always determined by the signs of sin θ and cos θ together (e.g. in QIII both are negative, so their quotient, tan, is positive).
Quick Review
- ASTC: Quadrant I all positive; II sin positive; III tan positive; IV cos positive.
- Each ratio's sign follows directly from the signs of x and y in that quadrant.
- tan θ's sign is the combined effect of sin θ's and cos θ's signs.