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 notes NDA 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

a\sin x+b\cos x

is a single sinusoid of amplitude

\sqrt{a^2+b^2}

. So its values run exactly over

[-\sqrt{a^2+b^2},\ \sqrt{a^2+b^2}]

— the maximum and minimum drop out immediately.

Definition

Write $a\sin x+b\cos x=R\sin(x+\varphi)$ with $R=\sqrt{a^2+b^2}$ . Then **max $=+\sqrt{a^2+b^2}$ , min $=-\sqrt{a^2+b^2}$ **. A constant $c$ added shifts the whole range to $[c-R,\ c+R]$ . The extremum is attained when $\sin(x+\varphi)=\pm 1$ .

Worked example

Find the maximum and minimum of

3\sin x+4\cos x

$R=\sqrt{3^2+4^2}=5$ .
Max $=5$ , min $=-5$ .

Answer:Maximum

5

, minimum

-5

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the maximum of

\sin\!\left(x+\tfrac\pi6\right)+\cos\!\left(x+\tfrac\pi6\right)

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Max of $a\sin x+b\cos x$ ?
2.
Min of $3\sin x+4\cos x$ ?
3.
Range of $2+\sin x+\cos x$ ?
4.
Max of $\cos x+\sqrt3\sin x$ ?

From the bank · past-year question

Example 1Trigonometric IdentitiesMODERATE

The maximum value of

\sin\!\left(x+\dfrac{\pi}{6}\right)+\cos\!\left(x+\dfrac{\pi}{6}\right)

in the interval

\left(0,\dfrac{\pi}{2}\right)

is attained at

[Q42 · Apr · 2017]

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^2\theta+n^2\tan^2\theta

\sec^2+\csc^2

combinations,

\cos+\sec

— AM-GM gives the minimum in one line, with equality pinning the optimal angle.

Definition

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

AM-GM minimum

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

Worked example

Find the minimum of

\cot^2\theta+9\tan^2\theta

AM-GM: $\cot^2\theta+9\tan^2\theta\ge 2\sqrt{9\cot^2\theta\tan^2\theta}=2\sqrt9=6$ .
Equality when $\cot^2\theta=9\tan^2\theta$ .

Answer:Minimum

=6

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the least value of

9\tan^2\theta+4\cot^2\theta

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Min of $\cot^2\theta+n^2\tan^2\theta$ ?
2.
Min of $\cos\theta+\sec\theta$ ( $\cos\theta>0$ )?
3.
Min of $\dfrac{a^2}{\cos^2x}+\dfrac{b^2}{\sin^2x}$ ?
4.
AM-GM equality holds when?

From the bank · past-year question

Example 2Trigonometric IdentitiesMODERATE

What is the least value of

25\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^2 x\in[0,1]

and the expression becomes a quadratic in

t

. Optimise the quadratic on

[0,1]

— vertex or endpoints. The same idea bounds a parameter via

\tan^2 A\ge 0

Definition

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

Worked example

Find the range of

A=\sin^2\theta+\cos^4\theta

Let $t=\sin^2\theta\in[0,1]$ : $A=t+(1-t)^2=t^2-t+1$ .
Vertex at $t=\tfrac12$ : $A_{\min}=\tfrac34$ ; endpoints $t=0,1$ give $A=1$ (max).

Answer:

A\in\left[\tfrac34,\ 1\right]

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the maximum of

3(\sin x-\cos x)^4+6(\sin x+\cos x)^2+4(\sin^6 x+\cos^6 x)

— outline the substitution.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Good substitution when only even powers appear?
2.
Min of $\sin^2\theta+\cos^4\theta$ ?
3.
After substituting, optimise the quadratic where?
4.
What bounds a parameter $K$ in these problems?

From the bank · past-year question

Example 3Trigonometric IdentitiesMODERATE

A = \sin^2\theta + \cos^4\theta

, then for all real

\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

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

Formulas (1)

AM-GM for reciprocal-type minima
AM-GM minimum
$u+v\ge 2\sqrt{uv}\ \ (u,v>0),\quad \text{equality at } u=v$

Mastery check — 5 interleaved questions

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

Example 1Trigonometric IdentitiesEASY

Consider the following for the items that follow: Let

x=\dfrac{\sin^{2}A+\sin A+1}{\sin A}

where

0<A\leq\dfrac{\pi}{2}

At what value of

A

does

x

attain the minimum value?

[Q38 · Apr · 2023]

Example 2Trigonometric IdentitiesMODERATE

Consider the following statements : 1.

\cos\theta + \sec\theta

can never be equal to

1\cdot5

. 2.

\tan\theta + \cot\theta

can never be less than 2. Which of the above statements is/are correct ?

[Q42 · Sep · 2019]

Example 3Trigonometric IdentitiesHARD

Directions for the following two (02) items: Read the following information and answer the two items that follow: Let

\frac{\tan 3A}{\tan A} = K

, where

\tan A \neq 0

and

K \neq \frac{1}{3}

For real values of

\tan A

K

\textbf{\text{cannot}}

lie between

[Q20 · Apr · 2020]

Example 4Trigonometric IdentitiesEASY

Let

p = \sin\alpha - \sin(\alpha - 90°)

What is the maximum value of

p

[Q35 · Apr · 2025]

Example 5Trigonometric IdentitiesMODERATE

Consider the following for the items that follow: Given that

m(\theta)=\cot^{2}\theta+n^{2}\tan^{2}\theta+2n

, where

n

is a fixed positive real number.

Under what condition does

m

attain the least value?

[Q54 · Apr · 2023]

Drill every past-year question on this subtopic

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

Drill the 22 questions

Drill every past-year question on this subtopic

Related notes