Trigonometry

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high importance~3 Q in Tier 119 formulas⚡ 11 shortcuts5 subtopics

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Subtopic 5 of 5·← Value-putting and given-ratio questions

Maximum and minimum values

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For $0^\circ\le\theta\le90^\circ$ :

$0\le\sin\theta,\cos\theta\le1$ ; $\sec\theta,\text{cosec}\theta\ge1$ ; $\tan\theta,\cot\theta$ are unbounded.
$\sin\theta+\cos\theta\in[1,\sqrt2]$ ; $\sin\theta-\cos\theta\in[-1,1]$ ; $\sin\theta\cos\theta\le\frac12$ .
$a\sin^2\theta+b\cos^2\theta$ has minimum $\min(a,b)$ and maximum $\max(a,b)$ .
$a\tan^2\theta+b\cot^2\theta\ (\text{and } a\sin^2+b\,\text{cosec}^2)$ has minimum $2\sqrt{ab}$ by AM–GM.
$\sec^2\theta+\text{cosec}^2\theta$ has minimum $2+2=4$ (at $45^\circ$ ).

📖 Notes 🧩 4 question types ⚡ Formulas 🎯 14 practice Q

Detailed notes

The amplitude formula (the workhorse)

For any angle $\theta$ : $-\sqrt{a^2 + b^2} \ \le\ a\sin\theta + b\cos\theta \ \le\ +\sqrt{a^2 + b^2}$ The maximum of $a\sin\theta + b\cos\theta$ is $\sqrt{a^2+b^2}$ and the minimum is its negative (unrestricted $\theta$ ); restricted to acute angles the range sits between the endpoint values — usually stated so the interior extremum is the answer. Instant values: $4\sin\theta + 3\cos\theta \to 5$ , $5\sin\theta + 12\cos\theta \to 13$ , $8\sin\theta - 15\cos\theta \to 17$ — Pythagorean pairs every time. On $0^\circ \le \theta \le 90^\circ$ : $\sin\theta + \cos\theta$ ranges from $1$ to $\sqrt2$ — max $\sqrt2$ at $45^\circ$ , min $1$ at the endpoints.

The product bound

$\sin\theta\cos\theta \le \frac12, \qquad 2\sin\theta\cos\theta = \sin 2\theta \le 1$ Both maxima occur at $\theta = 45^\circ$ . Any multiple $k\sin\theta\cos\theta$ peaks at $\frac{k}{2}$ .

Self-reciprocal sums (AM-GM)

$\tan^2\theta + \cot^2\theta \ge 2,\qquad \sec^2\theta + \cosec^2\theta \ge 4,\qquad \tan\theta + \cot\theta \ge 2$ All attained at $\theta = 45^\circ$ . The $\sec^2 + \cosec^2$ version: $\frac{1}{\sin^2\cos^2} \ge 4$ since $\sin^2\cos^2 \le \frac14$ . Any $x + \frac1x \ge 2$ shape with $x > 0$ follows the same rule, equality at $x = 1$ .

Ordering and comparison facts

On $(0^\circ, 90^\circ)$ : $\sin$ rises $0 \to 1$ , $\cos$ falls $1 \to 0$ , $\tan$ rises $0 \to \infty$ . So $\sin 60^\circ > \sin 45^\circ > \sin 30^\circ$ ; $\sin\theta > \cos\theta \iff \theta > 45^\circ$ ; $\tan\theta = \cot\theta$ only at $45^\circ$ . Also $\frac{1}{1+\sin^2\theta}$ is smallest when $\sin^2\theta$ is largest ( $= 1$ ), giving minimum $\frac12$ .

The 45° equality case and restricted domains

Every bound in this block is attained at $\theta = 45^\circ$ : the amplitude expressions reach $\pm\sqrt{a^2+b^2}$ only for unrestricted $\theta$ (the peak of $a\sin\theta + b\cos\theta$ sits at $\tan\theta = \frac{a}{b}$ , which is acute for positive $a, b$ — so on $0^\circ$ to $90^\circ$ the max $\sqrt{a^2+b^2}$ IS reachable, while the negative minimum is cut off at the endpoint $-b$ ). Always read the domain: "for all $\theta$ " allows the negative amplitude; " $0^\circ \le \theta \le 90^\circ$ " floors the minimum at the endpoint value. Options often include both — the domain decides.

Quick revision

$a\sin\theta + b\cos\theta$ : max $\sqrt{a^2+b^2}$ , min $-\sqrt{a^2+b^2}$ .
$\sin\theta\cos\theta \le \frac12$ ; $2\sin\theta\cos\theta \le 1$ , both at $45^\circ$ .
$\tan^2 + \cot^2 \ge 2$ ; $\sec^2 + \cosec^2 \ge 4$ ; $x + \frac1x \ge 2$ with equality at $x=1$ .
$\sin$ up, $\cos$ down on $(0^\circ, 90^\circ)$ ; crossover at $45^\circ$ .

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Max/min of a sinθ ± b cosθvery common4 practice Q

How to spot it:

'Find the maximum of 4sinθ + 3cosθ' — a linear combination with Pythagorean coefficients.

Max $= +\sqrt{a^2+b^2}$, min $= -\sqrt{a^2+b^2}$ for $a\sin\theta + b\cos\theta$.

Read off $(a, b)$ ; compute $\sqrt{a^2 + b^2}$ — expect a triplet.
Maximum positive, minimum negative (unless $\theta$ is restricted; then check endpoints).
Quote both if 'range' is asked.

Why: the expression is a single wave with amplitude $\sqrt{a^2+b^2}$ — the formula IS the answer.

Example: The maximum value of $4\sin\theta + 3\cos\theta$ is:

$\sqrt{4^2 + 3^2} = 5$ . Minimum: $-5$ (unrestricted $\theta$ ).

Type 2: Bounds of sinθ cosθcommon2 practice Q

How to spot it:

Max/min of sinθcosθ or 2sinθcosθ (sometimes with a coefficient or a plus-one).

$\sin\theta\cos\theta \le \frac12$; $2\sin\theta\cos\theta = \sin2\theta \le 1$; both peak at $45^\circ$.

Recognise the product form; halve the coefficient for the max of $k\sin\theta\cos\theta$ .
Or convert to $\sin 2\theta$ and use the wave range.
Minimum on acute angles is $0$ (endpoint); quote only if asked.

Why: the product is the double-angle formula in disguise — one conversion bounds it.

Example: The maximum value of $2\sin\theta\cos\theta$ ( $\theta$ acute) is:

$2\sin\theta\cos\theta = \sin 2\theta \le 1$ , attained at $\theta = 45^\circ$ .

Type 3: AM-GM self-reciprocal sumscommon3 practice Q

How to spot it:

Minimum of tan²+cot², sec²+cosec², tan+cot — or an equation whose sum equals the bound.

$x + \frac1x \ge 2$; $\sec^2\theta + \cosec^2\theta = \frac{1}{\sin^2\theta\cos^2\theta} \ge 4$.

Identify the self-reciprocal pair; quote the AM-GM bound.
$\sec^2 + \cosec^2$ : combine to $\frac{1}{\sin^2\cos^2}$ , then use $\sin\cos \le \frac12$ .
Equality case: $\theta = 45^\circ$ in every one of these.

Why: reciprocal pairs cannot be small simultaneously — the product $x \cdot \frac1x = 1$ forces the sum up.

Example: The minimum value of $\sec^2\theta + \cosec^2\theta$ is:

$= \frac{1}{\sin^2\theta\cos^2\theta} \ge \frac{1}{(1/2)^2} = 4$ , at $\theta = 45^\circ$ .

Type 4: Ordering and crossover comparisonscommon3 practice Q

How to spot it:

'Which is largest: sin 30, sin 45, sin 60?'; 'sinθ > cosθ for which θ?'; minimum of 1/(1+sin²θ).

On $(0^\circ, 90^\circ)$: sin increases, cos decreases; crossover at $45^\circ$; denominators largest → fraction smallest.

Monotonicity orders same-function comparisons instantly.
sin vs cos comparisons pivot at $45^\circ$ .
For $\frac{1}{1+\sin^2\theta}$ : minimise by maximising the denominator ( $\sin^2\theta = 1$ ).

Why: the monotone staircase of the three functions on the acute quadrant decides everything without computation.

Example: For $0^\circ < \theta < 90^\circ$ , $\sin\theta > \cos\theta$ holds when:

$\theta > 45^\circ$ — the crossover of the two graphs is at $45^\circ$ .

Formulas

Ranges

0\le\sin\theta\le1,\quad \sec\theta\ge1,\quad \sin\theta+\cos\theta\in[1,\sqrt2]

Weighted square form

a\sin^2\theta+b\cos^2\theta\in[\min(a,b),\max(a,b)]

AM-GM forms

a\tan^2\theta+b\cot^2\theta\ge2\sqrt{ab},\quad a\sin^2\theta+b\,\text{cosec}^2\theta\ge2\sqrt{ab}

Combined sec-cosec

\sec^2\theta+\text{cosec}^2\theta=2+\tan^2\theta+\cot^2\theta\ge4

Linear combination

a\sin\theta+b\cos\theta\text{ has maximum }\sqrt{a^2+b^2}

Shortcut tricks

⚡ Match the question to a range

Bounded pairs ( $\sin+\cos$ etc.) → memorised range. Weighted square sums → min of coefficients or $2\sqrt{ab}$ .

Example: Find the minimum value of $4\tan^2\theta+9\cot^2\theta$ for $0^\circ<\theta<90^\circ$ .

$2\sqrt{4\times9}=12$ (at $\tan^2\theta=\frac32$ ).

⚡ Weighted sin-cos squares are bounded by coefficients

$2\sin^2\theta+3\cos^2\theta=2+\cos^2\theta$ , so it ranges from 2 to 3 — min 2, max 3, no calculus.

Example: Find the minimum value of $5\sin^2\theta+12\cos^2\theta$ .

$=5+7\cos^2\theta\ge5$ ; minimum $5$ (at $\theta=0$ ).

Where students lose marks

AM-GM applied to $a\sin\theta+b\text{cosec}\theta$ where the equality point may be unreachable — for the square versions it is safe, check before using.
Quoting $\sqrt2$ as the minimum of $\sin+\cos$ — $\sqrt2$ is the maximum.
$\sec^2+\text{cosec}^2$ minimum quoted as 2 instead of 4.

Practice sets — 14 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 10 questions

Suggested time 6 min · wrong answers go to your mistake notebook automatically.