Trigonometry

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Maximum and minimum values

⏱ 3 min read🧩 5 question types🎯 14 practice Q

The idea in one minute

On zero to ninety degrees, sine and cosine stay between zero and one, while secant and cosecant stay at one or above. A sine-cosine combination peaks at the square root of a squared plus b squared, and reciprocal pairs bottom out at two by AM-GM.

The ranges

Function	Range on $0^\circ$ to $90^\circ$
$\sin\theta,\ \cos\theta$	$0$ to $1$
$\sec\theta,\ \cosec\theta$	$1$ or more
$\tan\theta,\ \cot\theta$	$0$ onwards, unbounded

$\sin0^\circ=0$ rises to $\sin90^\circ=1$ ; cosine runs the other way. They cross exactly once, at $45^\circ$ .

The table also explains why $\cosec\theta$ never lands between $0$ and $1$ : it is $\dfrac{1}{\sin\theta}$ and $\sin\theta\le1$ . The same logic lifts $\sec\theta$ to $1$ or above.

Rule: $\sin\theta>\cos\theta$ holds precisely when $\theta>45^\circ$ ; below it, cosine leads.

The amplitude rule

-\sqrt{a^2+b^2}\le a\sin\theta+b\cos\theta\le\sqrt{a^2+b^2}

$4\sin\theta+3\cos\theta$ peaks at $\sqrt{16+9}=5$ . The maximum of $a\sin+b\cos$ never needs calculus: square, add, root.

On the full circle the minimum is $-\sqrt{a^2+b^2}$ ; restricted to acute angles the low end is the smaller of $a$ and $b$ , hit at an endpoint. For $4\sin+3\cos$ on $0^\circ$ to $90^\circ$ the range is $[3,5]$ . Read the domain before answering.

Tip: The peak of $a\sin\theta+b\cos\theta$ answers half this subtopic. Compute $a^2+b^2$ first.

Products of sin and cos

\sin\theta\cos\theta=\frac{\sin2\theta}{2}\le\frac12

$2\sin\theta\cos\theta=\sin2\theta$ tops at $1$ , so $\sin\theta\cos\theta$ tops at $\dfrac12$ , both at $45^\circ$ .

Any question about 'maximum of sin times cos' is this line in disguise.

AM-GM floors

For positive $x$ : $x+\dfrac1x\ge2$ . So:

$\tan\theta+\cot\theta\ge2$ (product $1$ )
$\sec\theta+\cosec\theta\ge2$
$\sec^2\theta+\cosec^2\theta\ge4$ , since it equals $\dfrac{1}{\sin^2\theta\cos^2\theta}$ and $\sin^2\theta\cos^2\theta\le\dfrac14$

Each floor is attained where the two terms balance, at $45^\circ$ . Equality pins the angle exactly: $\tan\theta+\cot\theta=2$ forces $\tan\theta=1$ , hence $\theta=45^\circ$ . The floor and the balance point arrive together.

Watch: Secant and cosecant never fall below $1$ , so their sums have higher floors. Squares shift the floor again.

Weighted sin-squared and cos-squared

$5\sin^2\theta+12\cos^2\theta=5+7\cos^2\theta$ . Since $0\le\cos^2\theta\le1$ , the expression lives in $[5,12]$ : minimum $5$ , maximum $12$ .

The endpoints occur at $90^\circ$ (all sine) and $0^\circ$ (all cosine). Try $7\sin^2\theta+2\cos^2\theta=2+5\sin^2\theta$ , which lives in $[2,7]$ : the rewrite is mechanical whichever way the weights lean.

Remember: Rewrite the mix as one constant plus one squared term. The range reads off in one line, no differentiation.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common4 practice Q

Max and min of a sin plus b cos

How to spot it:

A linear mix of sine and cosine; an extreme asked.

Method

Compute $a^2+b^2$ .
Take the square root for the amplitude.
State max (and min if asked).

Why it works:

The amplitude formula replaces all calculus for this expression family.

Try this

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

Show solution

$\sqrt{16+9}=\sqrt{25}$ .
$=5$ .

Answer

Type 2common2 practice Q

Bounds of sin times cos

How to spot it:

A product or double of sin and cos; its maximum asked.

Method

Write the product as $\\dfrac{\\sin2\\theta}{2}$ .
Bound $\\sin2\\theta$ by $1$ .
Halve for the product.

Why it works:

The double-angle rewrite turns the product into a single bounded function.

Try this

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

Show solution

$2\sin\theta\cos\theta=\sin2\theta$ .
$\sin2\theta\le1$ .

Answer

Type 3common3 practice Q

AM-GM self-reciprocal sums

How to spot it:

A sum of a function and its reciprocal; a minimum asked.

Method

Confirm the product of the terms is $1$ .
Apply $x+\\dfrac1x\\ge2$ .
For squared versions, use $\\sin\\theta\\cos\\theta\\le\\dfrac12$ .

Why it works:

Reciprocal pairs always balance at their floor, which AM-GM locates instantly.

Try this

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

Show solution

$=\dfrac{1}{\sin^2\theta\cos^2\theta}$ .
$\sin\theta\cos\theta\le\dfrac12$ gives the minimum $4$ .

Answer

Type 4common3 practice Q

Ordering and crossover comparisons

How to spot it:

When is sin greater than cos, or tan greater than one.

Method

Place $45^\\circ$ as the crossover.
Below it cosine leads; above it sine leads.
Translate to the asked pair.

Why it works:

All such comparisons pivot on the single $45^\circ$ crossing, so one fact answers many questions.

Try this

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

Show solution

$\sin=\cos$ at $45^\circ$ .
Sine grows and cosine shrinks.
So $\\theta>45^\\circ$ .

Answer

theta > 45 degrees

Type 5common

Weighted sin-squared plus cos-squared range

How to spot it:

A weighted mix of squared sine and cosine; min and max asked.

Method

Rewrite as constant plus weight times one square.
Bound the square by $0$ and $1$ .
Read off both endpoints.

Why it works:

The rewrite makes the range a one-step reading exercise with no calculus.

Try this

The minimum and maximum of $5\sin^2\theta + 12\cos^2\theta$ ( $\theta$ acute) are:

Show solution

$=5+7\cos^2\theta$ .
$\cos^2\theta\in[0,1]$ , so the range is $[5,12]$ .

Answer

Minimum 5, maximum 12

Formula sheet

Amplitude of a sin + b cos

\max=\sqrt{a^2+b^2},\quad \min=-\sqrt{a^2+b^2}

General angles; on 0 to 90 check the endpoints too.

sin times cos

\sin\theta\cos\theta=\frac{\sin2\theta}{2}\le\frac12

Peak at 45 degrees.

AM-GM floor

x+\frac{1}{x}\ge2

For positive x; equality when x = 1.

Weighted squares

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

Rewrite as one constant plus one square.

Shortcuts that save time

⚡ Square, add, root

The maximum of a sin + b cos is the hypotenuse of the a-b right triangle. 4 and 3 give 5.

Example

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

Show solution

$\sqrt{a^2+b^2}=\sqrt{16+9}$ .
$=\sqrt{25}$ .
$=5$ .

Answer

⚡ AM-GM floor of two

Anything plus its own reciprocal bottoms at 2: tan + cot, sec + cosec, all the same.

Example

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

Show solution

$=\dfrac{1}{\sin^2\theta\cos^2\theta}$ and $\sin\theta\cos\theta\le\dfrac12$ .
So the minimum is $\dfrac{1}{(1/2)^2}=4$ .

Answer

⚡ Weighted squares: constant plus square

Rewrite 5 sin squared + 12 cos squared as 5 + 7 cos squared. The range is then obvious.

Example

The minimum value of $5\sin^2\theta + 12\cos^2\theta$ is:

Show solution

$=5+7\cos^2\theta$ .
$\cos^2\theta\ge0$ , so the minimum is $5$ .

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Taking the max of $4\sin+3\cos$ as $4+3=7$ .

The peak is $\sqrt{16+9}=5$ ; the functions never peak together.

Mistake 02

Saying $\sin\theta\cos\theta\le1$ .

The product tops at $\dfrac12$ at $45^\circ$ , since it equals $\dfrac{\sin2\theta}{2}$ .

Mistake 03

Giving the minimum of $\sec^2+\cosec^2$ as $2$ .

The floor is $4$ : the squares push it above the plain AM-GM pair.

Mistake 04

Claiming $\sin\theta>\cos\theta$ for all acute angles.

Only beyond $45^\circ$ ; below it cosine is larger.

Mistake 05

Weighted-square minimum taken at $45^\circ$ .

$5\sin^2+12\cos^2$ is smallest when the weight sits on the smaller value: $5$ at $90^\circ$ .

Quick revision

Read this the night before the exam.

Ranges: $\sin,\cos\in[0,1]$ ; $\sec,\cosec\ge1$ ; $\tan,\cot$ unbounded.
$\max(a\sin+b\cos)=\sqrt{a^2+b^2}$ ; square, add, root.
$\sin\theta\cos\theta\le\dfrac12$ ; $\sin2\theta\le1$ .
$x+\dfrac1x\ge2$ floors: $\tan+\cot\ge2$ , $\sec^2+\cosec^2\ge4$ .
Weighted squares: rewrite as constant plus square, read the range.
$\sin=\cos$ only at $45^\circ$ ; that is the ordering crossover.

Practice: 14 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

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

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