Ratios and Graphs of Trigonometric Functions · 6.1.3
Finding angles from trig ratios
Given a value of sin θ, cos θ or tan θ (positive or negative), students find all angles θ between 0° and 360° that produce it. This uses the reference angle from a calculator together with the quadrant rule (ASTC), since each ratio value usually corresponds to two angles within one full turn.
The official learning standard (6.1.3)
“Determine the angle when the value of sine, cosine and tangent are given.”
What it means
Given a value of sin θ, cos θ or tan θ (positive or negative), students find all angles θ between 0° and 360° that produce it. This uses the reference angle from a calculator together with the quadrant rule (ASTC), since each ratio value usually corresponds to two angles within one full turn.
How it is examined
In Paper 1, students calculate an angle directly from a given ratio using a calculator and the quadrant rule, often as a standalone multiple-choice or short-answer item. In Paper 2, this skill is usually embedded within a larger question, for example alongside triangle problems, trigonometric graphs, or equations that first require finding a ratio value before the angle.
Worked example
Given that cos θ = -0.5, find the values of θ in the range 0° ≤ θ ≤ 360°.
- Find the reference angle: cos⁻¹(0.5) = 60°.
- Since cos θ is negative, θ lies in Quadrant II or Quadrant III.
- Quadrant II: θ = 180° - 60° = 120°.
- Quadrant III: θ = 180° + 60° = 240°.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I know which quadrants to use?
Look at the sign of the given ratio and the ASTC rule: All ratios are positive in Quadrant I, Sine only in II, Tangent only in III, and Cosine only in IV. Match the sign of your value to the quadrants where that ratio is positive.
Why are there usually two answers?
Within one full turn (0° to 360°), each sine, cosine or tangent value (except at special boundary angles) occurs in exactly two quadrants with the same magnitude, giving two possible angles. Always check both quadrants indicated by the sign of the ratio before stating your final answers.
What if tan θ is negative, which quadrants apply?
Tangent is negative in Quadrant II and Quadrant IV (since it is positive only in I and III). Find the reference angle using the positive value on your calculator, then compute θ = 180° - reference angle for Quadrant II and θ = 360° - reference angle for Quadrant IV.