LearningMath
🌲

The Trigonometry Decision Tree

The Paper 1 & 2 engine — instead of memorising a wall of identities, use this "if you see X → do Y" flowchart to eliminate exam hesitation the moment you see a trig question.

Simplifying & proving identities

If you see cos⁡(2θ)\cos(2\theta) with only sines in the rest of the equation

1−2sin⁡2θ1 - 2\sin^2\theta

If you see cos⁡(2θ)\cos(2\theta) with only cosines in the rest of the equation

2cos⁡2θ−12\cos^2\theta - 1

If you see fractions, or need to factorise a difference of squares

cos⁡2θ−sin⁡2θ\cos^2\theta - \sin^2\theta

Co-functions & reduction formulae

If the angle contains 90∘ or 270∘90^\circ \text{ or } 270^\circ

(e.g. sin(90° + θ)) — switch the function (sin ↔ cos) and determine the quadrant sign.

sin⁡(90∘+θ)→cos⁡θ\sin(90^\circ + \theta) \rightarrow \cos\theta

If the angle is negative (e.g. cos⁡(−θ))(\text{e.g. } \cos(-\theta))

cos "swallows" the negative sign, sin pushes it out.

cos⁡(−θ)=cos⁡θ,sin⁡(−θ)=−sin⁡θ\cos(-\theta) = \cos\theta, \quad \sin(-\theta) = -\sin\theta

General solutions (trig equations)

  1. 1. Isolate the trig termsin⁡θ=k\sin\theta = k
  2. 2. Find the reference angleθref=sin⁡−1(k)\theta_{ref} = \sin^{-1}(k)
  3. 3. Apply the quadrant ruleθ=180∘−θref  (or relevant quadrant)\theta = 180^\circ - \theta_{ref} \;(\text{or relevant quadrant})
  4. 4. Add the general termθ=…+k⋅360∘  (or k⋅180∘ for tan⁡)\theta = \ldots + k \cdot 360^\circ \;(\text{or } k \cdot 180^\circ \text{ for } \tan)

Want more of these?

Check the interactive Grade 12 formula reference for every other CAPS formula, with exam-trap warnings built in.

Open the Formula Rosetta Stone

← Back to all guides