1. Understanding Quadrants in the Coordinate Plane
The coordinate plane is divided into four quadrants by the x-axis and y-axis. When an angle is measured from the positive x-axis, its terminal side falls in one of these quadrants.
- Quadrant I: both x and y are positive
- Quadrant II: x is negative, y is positive
- Quadrant III: both x and y are negative
- Quadrant IV: x is positive, y is negative
Trigonometric ratios depend on the signs of x and y, so they change with the quadrant.
2. ASTC Rule (All Students Take Coffee)
An easy memory trick called the ASTC Rule tells you which trigonometric ratios are positive in each quadrant.
- A (All) → In Quadrant I, all trigonometric ratios are positive.
- S (Students) → In Quadrant II, sin and cosec are positive.
- T (Take) → In Quadrant III, tan and cot are positive.
- C (Coffee) → In Quadrant IV, cos and sec are positive.
3. Sign Table for All Trigonometric Ratios
The following table summarises which ratios are positive or negative in each quadrant.
| Quadrant | sin | cos | tan | cot | sec | cosec |
|---|---|---|---|---|---|---|
| I | + | + | + | + | + | + |
| II | + | − | − | − | − | + |
| III | − | − | + | + | − | − |
| IV | − | + | − | − | + | − |
4. Why Signs Change in Each Quadrant
Trigonometric ratios relate the sides of a right triangle formed inside each quadrant. Using coordinate geometry:
- sin corresponds to y-coordinate → positive if y>0, negative if y<0
- cos corresponds to x-coordinate → positive if x>0, negative if x<0
- tan = y/x, so its sign depends on signs of both x and y
Hence the quadrant naturally decides the positive or negative sign of each ratio.
5. Examples to Understand the Signs
Example 1: If an angle lies in Quadrant II, then:
- cos θ is negative
- sin θ is positive
- tan θ is negative
Example 2: If tan θ is positive but sin θ is negative, the angle must be in Quadrant III (since both sin and cos are negative there).