IP AMaths Notes (Upper Sec, Year 3-4): 13) Trigonometry II

Study guide08 Nov 2025, 00:00 ZUpdated 30 Nov 2025

Trig equations, compound angles, and auxiliary angle method for IP AMaths.

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Q: What does IP AMaths Notes (Upper Sec, Year 3-4): 13) Trigonometry II cover?
A: Trig equations, compound angles, and auxiliary angle method for IP AMaths.

With radian fluency in hand, solve trig equations by mapping solutions across quadrants and applying compound-angle identities.

Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.

New to the Integrated Programme? Start with What is IP? | Browse all free IP notes.

This topic stays within the SEAB GCE O-Level Additional Mathematics (4049) scope: compound-angle identities, auxiliary-angle rewriting, and solving trigonometric equations in radians.

Status: SEAB O-Level Additional Mathematics 4049 syllabus (exams from 2025) checked 2025-11-30 - scope unchanged; remains the reference for this note.

The core idea is simple: Trigonometry II is about solving equations across the correct interval.

Use it as a working check: Reduce the equation to a familiar trig function, find all quadrant solutions, then check the requested range in radians.

Then go one layer deeper: Example: if sin x = 1/2 from 0 to just before 2pi, include both pi/6 and 5pi/6, not just the calculator's first answer.

Solving route for trig equations

The goal is not just to solve one angle. It is to find every angle in the stated interval.

Question cue	First move	Check before final answer
Quadratic in $\sin x$ , $\cos x$

Reviewed by

Marcus Pang·Managing Director (Maths)

Sources

SEAB GCE O-Level Additional Mathematics (4049) syllabus (examinations from 2025)

Equation shape	Safer first move	Why it works	Trap to avoid
Product form, such as $\cos x(2\sin x - 1) = 0$	Set each factor to zero.	Each factor can create a separate family of angles.	Dividing by $\cos x$ and losing $\cos x = 0$ solutions.
Double-angle form, such as $\sin 2x = \cos x$	Expand if a common factor appears.	Expansion can reveal a factorisation route.	Solving only for $2x$ and forgetting the original interval for $x$ .
Linear combination, such as $3\sin x + 4\cos x = 2$	Convert to $R\sin(x+\alpha)$ or $R\cos(x-\alpha)$
Quadratic in one trig function	Substitute temporarily, then convert back.	The algebra step is separate from the angle step.	Accepting impossible roots such as $\sin x = 2$ .

Step	What to write	Common trap
Shift the interval	If $0 \leq x \lt 2\pi$ , then $\alpha \leq x+\alpha \lt 2\pi+\alpha$ .	Solving only in $0 \leq x+\alpha \lt 2\pi$ and missing an endpoint solution.
Solve the sine or cosine equation	List every value of $x+\alpha$ inside the shifted interval.	Keeping only the calculator's principal angle.
Subtract $\alpha$	Convert each whole-angle solution back to $x$ .	Forgetting to subtract $\alpha$ from every branch.
Check the original interval	Keep only answers that satisfy the original range for $x$ .	Leaving a negative value such as $-0.516$ without adding $2\pi$ when the range is $0 \leq x \lt 2\pi$ .

IP AMaths Notes (Upper Sec, Year 3-4): 13) Trigonometry II

Solving route for trig equations

Sources

Equation route checkpoint

1 Identities to deploy

2 Worked example - Solve trig equation

3 Worked example - Auxiliary angle

Auxiliary-angle interval checkpoint

4 Worked example - Using double-angle identities in equations

5 Practice Quiz

6 Try this

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