Ratios and Graphs of Trigonometric Functions · 6.1.1
Trig ratios in quadrants II–IV
Using the unit circle, students investigate how sin, cos and tan of an angle greater than 90° relate to the same ratio of its acute reference angle, including which ratios are positive or negative in each quadrant, then form and verify a general conjecture.
The official learning standard (6.1.1)
“Make and verify conjecture about the value of sine, cosine and tangent for angles in quadrants II, III and IV with the corresponding reference angle.”
What it means
Using the unit circle, students investigate how sin, cos and tan of an angle greater than 90° relate to the same ratio of its acute reference angle, including which ratios are positive or negative in each quadrant, then form and verify a general conjecture.
How it is examined
Paper 1 frequently tests this directly as short objective items, finding sin/cos/tan of an obtuse or reflex angle using its reference angle, or identifying its sign. Paper 2 uses the concept within larger trigonometric equation or graph-sketching questions.
Worked example
Using the reference angle, express sin150°, cos150° and tan150° in terms of the ratio of the reference angle, and hence find their values.
- 150° lies in quadrant II (90° < 150° < 180°); the reference angle is 180° − 150° = 30°.
- In quadrant II, sine is positive, cosine is negative and tangent is negative.
- sin150° = sin30° = 1/2 (positive, matches quadrant II).
- cos150° = −cos30° = −√3/2 (negative, matches quadrant II).
- tan150° = −tan30° = −√3/3 (negative, matches quadrant II).
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What is a reference angle?
The reference angle is the acute angle, between 0° and 90°, formed between the terminal arm of the given angle and the x-axis. It gives the magnitude of the trig ratio, while the quadrant determines whether the value is positive or negative.
How do I know if a ratio is positive or negative in a given quadrant?
Use the ASTC rule: All ratios are positive in quadrant I, only Sine is positive in quadrant II, only Tangent is positive in quadrant III, and only Cosine is positive in quadrant IV.
How do I find the reference angle for angles in each quadrant?
For an angle θ measured from the positive x-axis: in quadrant II use (180° − θ), in quadrant III use (θ − 180°), and in quadrant IV use (360° − θ).