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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.

Test yourself — a quick 14-question Trigonometric Equations recall quiz, instant score.

Concept 1 of 2: Solving Two Equations Together

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.
  • The quadrant rule that does the pinning (ASTC): in quadrant I All ratios are positive; in II only Sine (and cosec); in III only Tangent (and cot); in IV only Cosine (and sec). So cot⁡θ=−3\cot\theta=-\sqrt3 puts θ\theta in II or IV, csc⁡θ=−2\csc\theta=-2 puts it in III or IV — together, quadrant IV, and with reference angle π6\tfrac\pi6 the answer on [0,2π)[0,2\pi) is 11π6\tfrac{11\pi}{6}. Place the reference angle as π−α\pi-\alpha, π+α\pi+\alpha, 2π−α2\pi-\alpha for quadrants II, III, IV.

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.

The same idea in a real exam question:

NDA · Sep 2025 · Q47Moderate

Example 1 · Trigonometric Equations · Simultaneous and Combined Trigonometric Systems

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?
Drill 3 more on solving two equations together

Concept 2 of 2: Reducing a Combined System

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.

The same idea in a real exam question:

NDA · Apr 2020 · Q24Hard

Example 2 · Trigonometric Equations · Simultaneous and Combined Trigonometric Systems

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)?
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

Test yourself on Trigonometric Equations

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