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NDA Maths · Trigonometric Equations

Simultaneous & Combined Trigonometric Systems

Two trig equations in the same angle (or two angles) are solved together: find each equation's solution set and intersect them, or combine the equations into one with a substitution.

All Trigonometric Equations notesNDA Maths strategy

Why this matters

7 PYQs, 3 HARD. The reliable approach is to solve each condition for its own solution set and keep only the common values — or, for a combined system, introduce a substitution like s = sin x + cos x that fuses both equations into a single solvable one.

Concept 1 of 2

Solving Two Equations Together

Intuition

When θ must satisfy two trig conditions at once, solve each separately for its list of angles in the interval, then take ONLY the values that appear in both lists. A pair of conditions usually pins the quadrant and leaves one common solution.

Definition

  • Intersect the solution sets: for cot⁡θ=−3\cot\theta = -\sqrt3 AND csc⁡θ=−2\csc\theta = -2, list each in the interval and keep the common angle (the two conditions fix both the reference angle and the quadrant).
  • Two-angle systems: sin⁡(A+B)=1\sin(A+B) = 1 and 2sin⁡(A−B)=12\sin(A-B) = 1 give A+B=π2A+B = \tfrac{\pi}{2}, A−B=π6A-B = \tfrac{\pi}{6}; solve the linear pair for A,BA, B, then any required ratio.
  • **sin⁡α+sin⁡β=0=cos⁡α+cos⁡β\sin\alpha + \sin\beta = 0 = \cos\alpha + \cos\beta:** both are negated, forcing α=π+β\alpha = \pi + \beta.

Common solution = intersection

{θ:eqn 1}∩{θ:eqn 2}\{\theta : \text{eqn 1}\} \cap \{\theta : \text{eqn 2}\}

Worked example

If sin⁡(A+B)=1\sin(A+B) = 1 and sin⁡(A−B)=12\sin(A-B) = \tfrac12 with A,BA, B acute, find AA and BB.
  1. sin⁡(A+B)=1⇒A+B=π2\sin(A+B) = 1 \Rightarrow A+B = \tfrac{\pi}{2}; sin⁡(A−B)=12⇒A−B=π6\sin(A-B) = \tfrac12 \Rightarrow A-B = \tfrac{\pi}{6}.
  2. Add and subtract: 2A=π2+π6=2π32A = \tfrac{\pi}{2} + \tfrac{\pi}{6} = \tfrac{2\pi}{3}, so A=π3A = \tfrac{\pi}{3}, B=π6B = \tfrac{\pi}{6}.
Answer:A=60∘, B=30∘A = 60^\circ,\ B = 30^\circ.

From the bank · past-year question

Example 1Trigonometric EquationsMODERATE
How many values of θ\theta, where −π<θ<π-\pi<\theta<\pi, satisfy both the equations cot⁡θ=−3\cot\theta=-\sqrt{3} and csc⁡θ=−2\csc\theta=-2 simultaneously?

[Q47 · Sep · 2025]

Drill 3 more on solving two equations together

Concept 2 of 2

Reducing a Combined System

Intuition

Some systems mix sin x + cos x, tan x + cot x, sec x + csc x in one equation. A single substitution, s = sin x + cos x, expresses all of them (since sin x cos x = (s²−1)/2), collapsing the tangle into one quadratic in s.

Definition

  • The s = sin x + cos x substitution: then sin⁡xcos⁡x=s2−12\sin x\cos x = \tfrac{s^2-1}{2}, sin⁡2x=s2−1\sin 2x = s^2 - 1, tan⁡x+cot⁡x=2sin⁡2x\tan x + \cot x = \dfrac{2}{\sin 2x}, sec⁡x+csc⁡x=2ssin⁡2x\sec x + \csc x = \dfrac{2s}{\sin 2x}. The whole equation becomes a polynomial in ss.
  • Eliminate between two relations: for cos⁡2B=3sin⁡2A\cos 2B = 3\sin^2 A and 3sin⁡2A=2sin⁡2B3\sin 2A = 2\sin 2B, substitute one into the other to reach a clean angle relation such as A+2B=π2A + 2B = \tfrac{\pi}{2}.

The s-substitution

s=sin⁡x+cos⁡x ⇒ sin⁡xcos⁡x=s2−12,sin⁡2x=s2−1s = \sin x + \cos x \ \Rightarrow\ \sin x\cos x = \tfrac{s^2-1}{2},\quad \sin 2x = s^2 - 1

Worked example

If sin⁡x+cos⁡x=12\sin x + \cos x = \tfrac{1}{2}, find sin⁡2x\sin 2x.
  1. Square: (sin⁡x+cos⁡x)2=14(\sin x + \cos x)^2 = \tfrac14, i.e. 1+sin⁡2x=141 + \sin 2x = \tfrac14.
  2. sin⁡2x=14−1\sin 2x = \tfrac14 - 1.
Answer:sin⁡2x=−34\sin 2x = -\dfrac{3}{4}.

From the bank · past-year question

Example 2Trigonometric EquationsHARD
A and B are positive acute angles such that cos⁡2B=3sin⁡2A\cos 2B = 3\sin^2 A and 3sin⁡2A=2sin⁡2B3\sin 2A = 2\sin 2B. What is the value of (A+2B)(A+2B)?

[Q24 · Apr · 2020]

Drill 2 more on reducing a combined system

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Solving Two Equations Together

    Common solution = intersection

    {θ:eqn 1}∩{θ:eqn 2}\{\theta : \text{eqn 1}\} \cap \{\theta : \text{eqn 2}\}
  • Reducing a Combined System

    The s-substitution

    s=sin⁡x+cos⁡x ⇒ sin⁡xcos⁡x=s2−12,sin⁡2x=s2−1s = \sin x + \cos x \ \Rightarrow\ \sin x\cos x = \tfrac{s^2-1}{2},\quad \sin 2x = s^2 - 1

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Trigonometric EquationsEASY
If sin⁡(A+B)=1\sin(A+B)=1 and 2sin⁡(A−B)=12\sin(A-B)=1, where 0<A,B<π20<A,B<\frac{\pi}{2}, then what is tan⁡A:tan⁡B\tan A:\tan B equal to?

[Q24 · Sep · 2021]

Example 2Trigonometric EquationsHARD
Consider the following for the items that follow: Given that sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+cosec⁡x=7\sin x+\cos x+\tan x+\cot x+\sec x+\cosec x=7.
The given equation can be reduced to

[Q47 · Apr · 2023]

Example 3Trigonometric EquationsMODERATE
If sin⁡α+sin⁡β=0=cos⁡α+cos⁡β\sin\alpha + \sin\beta = 0 = \cos\alpha + \cos\beta, where 0<β<α<2π0 < \beta < \alpha < 2\pi, then which is correct?

[Q47 · Apr · 2018]

Example 4Trigonometric EquationsHARD
Consider the following for the items that follow: Given that sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+cosec⁡x=7\sin x+\cos x+\tan x+\cot x+\sec x+\cosec x=7.
If sin⁡2x=a−bc\sin2x=a-b\sqrt{c}, where aa and bb are natural numbers and cc is prime number, then what is the value of a−b+2ca-b+2c?

[Q48 · Apr · 2023]

Example 5Trigonometric EquationsEASY
If 2tan⁡A=3tan⁡B=12\tan A = 3\tan B = 1, then what is tan⁡(A−B)\tan(A-B) equal to ?

[Q65 · Sep · 2019]

Drill every past-year question on this subtopic

7 questions from the bank — paginated, with cart and Word-export support.

Drill the 7 questions