Ratios and Graphs of Trigonometric Functions · Form 5
Ratios and Graphs of Trigonometric Functions: Revision Notes
A tight revision summary of Ratios and Graphs of Trigonometric Functions for SPM Mathematics, the key ideas, the formulae the exam gives you, and what to focus on before the paper.
The big idea
You already know sine, cosine and tangent for angles in a right-angled triangle. This chapter extends them to any angle from 0° to 360°, teaches you to find their values (including negative ones), and to draw and read the wave-like graphs of these functions.
Key ideas to revise
- Values beyond 90°. Past 90° the ratios can be negative. Knowing which are positive in each quadrant prevents most sign errors.
- The shape of each graph. Sine and cosine are smooth waves; tangent repeats with breaks. Recognising the shape helps you sketch and read them.
- Solving from a graph. Reading where a trig graph meets a given value is a common way the chapter is examined.
Each key idea as a quick worked example
- Values beyond 90° (quadrant signs): find cos 240°. Reference angle = 240° − 180° = 60°. In the 3rd quadrant cosine is negative, so cos 240° = −cos 60° = −1/2.
- Shape of each graph: sketch y = tan x for 0° ≤ x ≤ 360°. It crosses zero at 0°, 180° and 360°, has vertical asymptotes at 90° and 270°, and repeats every 180°.
- Solving from a graph: to solve tan x = 1 for 0° ≤ x ≤ 360°, the reference angle is 45°; tangent is positive in the 1st and 3rd quadrants, so x = 45° and x = 225°.
Your pre-paper checklist for this chapter
- Re-derive the quadrant sign diagram (CAST): all positive in Q1, only sine in Q2, only tangent in Q3, only cosine in Q4.
- Re-derive the special-angle values from the two set-squares: the 30°–60°–90° triangle gives sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; the 45°–45°–90° triangle gives sin 45° = cos 45° = 1/√2.
- What the paper gives you: the range (for example 0° ≤ x ≤ 360°) and the equation, but NOT the trig values, so the special angles must be memorised.
- The one habit that saves marks: find the reference angle first, place the signs by quadrant, then list every solution inside the stated range before writing the final line.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How many cycles of y = sin x should appear between 0° and 360°?
Exactly one complete cycle. The curve of y = sin x starts at 0, rises to its peak of +1 at 90°, returns to 0 at 180°, falls to its trough of −1 at 270°, and climbs back to 0 at 360°.
If you have drawn two humps in that range you have accidentally doubled the frequency.
Do I need to memorise the trig values or are they on the formula sheet?
No. The official SPM formula sheet does not list trigonometric values, so the special angles must be in your memory.
Learn the 30°–60°–90° and 45°–45°–90° triangles; from those two set-squares you can rebuild sin, cos and tan of 30°, 45° and 60° exactly, and read 0°, 90° and 180° straight off the graph.
What is the quickest way to check a value like cos 240° = −1/2?
Two quick checks. First, reason it out: 240° is in the third quadrant where cosine is negative, and its reference angle 60° gives cos 60° = 1/2, so the value is −1/2.
Second, confirm on a calculator set to degree mode, not radians. If the two disagree, your mode is wrong.