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NDA Maths · Trigonometric Identities

Maximum & Minimum Values

Three reliable tools: the a·sinx + b·cosx amplitude bound, AM-GM for reciprocal sums, and substitution to a quadratic in sin²x — covering almost every extremum question.

All Trigonometric Identities notesNDA Maths strategy

Why this matters

Optimisation questions look varied but reduce to one of three moves. Knowing which to reach for — amplitude √(a²+b²), AM-GM, or a quadratic substitution — turns a scary-looking max/min into a one-liner.

Concept 1 of 3

Range of a·sin x + b·cos x

Intuition

Any combination asin⁡x+bcos⁡xa\sin x+b\cos x is a single sinusoid of amplitude a2+b2\sqrt{a^2+b^2}. So its values run exactly over [−a2+b2, a2+b2][-\sqrt{a^2+b^2},\ \sqrt{a^2+b^2}] — the maximum and minimum drop out immediately.

Definition

Write asin⁡x+bcos⁡x=Rsin⁡(x+φ)a\sin x+b\cos x=R\sin(x+\varphi) with R=a2+b2R=\sqrt{a^2+b^2}. Then **max =+a2+b2=+\sqrt{a^2+b^2}, min =−a2+b2=-\sqrt{a^2+b^2}**. A constant cc added shifts the whole range to [c−R, c+R][c-R,\ c+R]. The extremum is attained when sin⁡(x+φ)=±1\sin(x+\varphi)=\pm 1.

max = +√(a²+b²) = 5min = −√(a²+b²) = −53 sin x + 4 cos x

Worked example

Find the maximum and minimum of 3sin⁡x+4cos⁡x3\sin x+4\cos x.
Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 1Trigonometric IdentitiesMODERATE
The maximum value of sin⁡ ⁣(x+π6)+cos⁡ ⁣(x+π6)\sin\!\left(x+\dfrac{\pi}{6}\right)+\cos\!\left(x+\dfrac{\pi}{6}\right) in the interval (0,π2)\left(0,\dfrac{\pi}{2}\right) is attained at

[Q42 · Apr · 2017]

Max of asin⁡x+bcos⁡xa\sin x+b\cos x is a2+b2\sqrt{a^2+b^2}, NOT a+ba+b

Because sin⁡x\sin x and cos⁡x\cos x hit 11 at *different* values of xx, you cannot add their maxima — the peak is a2+b2\sqrt{a^2+b^2}, reached when the single sinusoid Rsin⁡(x+φ)R\sin(x+\varphi) equals 11. For 3sin⁡x+4cos⁡x3\sin x+4\cos x the maximum is 9+16=5\sqrt{9+16}=5, not 3+4=73+4=7. **Whenever you see asin⁡x+bcos⁡xa\sin x+b\cos x, reach for the amplitude a2+b2\sqrt{a^2+b^2}, never the coefficient sum.**
Drill 10 more on range of a·sin x + b·cos x

Concept 2 of 3

AM-GM for reciprocal-type minima

Intuition

When an expression is a sum of a term and (a constant times) its reciprocal — cot⁡2θ+n2tan⁡2θ\cot^2\theta+n^2\tan^2\theta, sec⁡2+csc⁡2\sec^2+\csc^2 combinations, cos⁡+sec⁡\cos+\sec — AM-GM gives the minimum in one line, with equality pinning the optimal angle.

Definition

By AM-GM, u+v≥2uvu+v\ge 2\sqrt{uv} for positive u,vu,v, equality at u=vu=v. So cot⁡2θ+n2tan⁡2θ≥2n\cot^2\theta+n^2\tan^2\theta\ge 2n, and cos⁡θ+sec⁡θ≥2\cos\theta+\sec\theta\ge 2. For a2cos⁡2x+b2sin⁡2x\dfrac{a^2}{\cos^2 x}+\dfrac{b^2}{\sin^2 x}, the minimum is (a+b)2(a+b)^2 (Cauchy–Schwarz / AM-GM).

AM-GM minimum

u+v≥2uv  (u,v>0),equality at u=vu+v\ge 2\sqrt{uv}\ \ (u,v>0),\quad \text{equality at } u=v

Worked example

Find the minimum of cot⁡2θ+9tan⁡2θ\cot^2\theta+9\tan^2\theta.
Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 2Trigonometric IdentitiesMODERATE
What is the least value of 25csc⁡2x+36sec⁡2x25\csc^2 x + 36\sec^2 x?

[Q26 · Apr · 2019]

Drill 6 more on am-gm for reciprocal-type minima

Concept 3 of 3

Substitute to a quadratic (let t = sin²x)

Intuition

When sin and cos appear only as even powers, set t=sin⁡2x∈[0,1]t=\sin^2 x\in[0,1] and the expression becomes a quadratic in tt. Optimise the quadratic on [0,1][0,1] — vertex or endpoints. The same idea bounds a parameter via tan⁡2A≥0\tan^2 A\ge 0.

Definition

Substitute t=sin⁡2xt=\sin^2 x (so cos⁡2x=1−t\cos^2 x=1-t, t∈[0,1]t\in[0,1]), reduce to f(t)=αt2+βt+γf(t)=\alpha t^2+\beta t+\gamma, and read off the extremum at the vertex t=−β2αt=-\tfrac{\beta}{2\alpha} (if in range) or at an endpoint. For parameter questions, tan⁡2A=g(K)≥0\tan^2 A=g(K)\ge 0 constrains the allowed KK.

Worked example

Find the range of A=sin⁡2θ+cos⁡4θA=\sin^2\theta+\cos^4\theta.
Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 3Trigonometric IdentitiesMODERATE
If A=sin⁡2θ+cos⁡4θA = \sin^2\theta + \cos^4\theta, then for all real θ\theta, which one of the following is correct?

[Q50 · Sep · 2018]

Drill 3 more on substitute to a quadratic (let t = sin²x)

Summary — formulas & gotchas at a glance

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Formulas (1)

  • AM-GM for reciprocal-type minima

    AM-GM minimum

    u+v≥2uv  (u,v>0),equality at u=vu+v\ge 2\sqrt{uv}\ \ (u,v>0),\quad \text{equality at } u=v

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