Ratios and Graphs of Trigonometric Functions · Form 5
Why does the quadrant decide the sign?
Once an angle is bigger than 90° we place it on a circle of radius 1, where cos θ is the x-coordinate and sin θ is the y-coordinate. Because x and y can be positive or negative in different parts of the circle, the sign of each ratio depends on which quadrant the angle lands in.
From triangle to circle
A right-angled triangle only has angles up to 90°, so to make sense of 120° or 250° we draw the angle from the positive x-axis on a unit circle centred at the origin. The point where the angle's arm meets the circle has coordinates (cos θ, sin θ), and tan θ is simply sin θ ÷ cos θ, or y ÷ x.
As the arm sweeps around, those coordinates move through positive and negative regions, and the ratios follow.
The CAST pattern
Split the circle into four quadrants of 90° each. In the first (0°–90°) all three ratios are positive; in the second (90°–180°) only sine is positive; in the third (180°–270°) only tangent is positive; in the fourth (270°–360°) only cosine is positive.
Students often memorise this as CAST or ASTC, one letter for the ratio that stays positive in each quadrant. Making this pattern automatic removes the single biggest source of sign errors in the whole chapter.
The misconception
Many students think an angle like 210° is meaningless because they cannot draw that triangle, or they assume every ratio must be positive. In fact the size of the ratio comes from the related acute angle, and the quadrant only decides the + or − sign in front.
For example sin 150° = +0.5 while sin 210° = −0.5: the same size, but opposite signs because they sit in different quadrants.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I know if sin, cos or tan is positive or negative in a given quadrant?
Use "All Silver Tea Cups" (ASTC): in Quadrant I all ratios are positive, II only sine, III only tangent, IV only cosine. This comes from the x (cos) and y (sin) coordinates on the unit circle changing sign as the angle moves past 90°, 180° and 270°.
Why do we use a unit circle instead of a triangle for angles above 90°?
A right-angled triangle only works for angles between 0° and 90°, but the unit circle lets any angle be placed around a full rotation. Reading cos θ as the x-coordinate and sin θ as the y-coordinate extends the ratios naturally to obtuse and reflex angles.
What's a common mistake when finding the reference angle in a different quadrant?
Students forget to subtract from the correct baseline, for Quadrant II use 180° − θ, Quadrant III use θ − 180°, and Quadrant IV use 360° − θ. Using the wrong formula gives the correct ratio value but the wrong sign, or an angle in the wrong quadrant entirely.