Trigonometric substitution — x = a sin θ, a tan θ, a sec θ cases: Foundations

Before you can even read the parent note, a whole stack of symbols and pictures has to be solid. This page builds every one of them from nothing, in the order they lean on each other. If a single link below feels shaky, that is exactly the piece to slow down on.

0. The map you are building towards

So the later sections have something concrete to point at, here is the whole apparatus of the parent topic in one table. You are not expected to understand it yet — every symbol in it is defined below. Keep it handy; sections 3, 7 and 9 all refer back to specific columns.

Definition The three substitutions at a glance

Radical	Substitution	$d x$	Root becomes	Domain of $x$	Range of $θ$
$a^{2} - x^{2}$	$x = a sin θ$	$a cos θ d θ$	$a cos θ$	$∣ x ∣ \leq a$	$θ \in [- \frac{π}{2}, \frac{π}{2}]$
$a^{2} + x^{2}$	$x = a tan θ$	$a sec^{2} θ d θ$	$a sec θ$	all real $x$	$θ \in (- \frac{π}{2}, \frac{π}{2})$
$x^{2} - a^{2}$	$x = a sec θ$	$a sec θ tan θ d θ$	$a ∣ tan θ ∣$	$∣ x ∣ \geq a$	$θ \in [0, π] ∖ {\frac{π}{2}}$

Here $a > 0$ . Each domain of $x$ column says when that triangle model even applies; each range of $θ$ column is the interval that makes the root come out $\geq 0$ . Both are derived, not decreed — see sections 3 and 7.

1. The right triangle — the picture underneath everything

Everything in this topic lives on ONE picture: a right triangle.

Figure — Trigonometric substitution — x = a sin θ, a tan θ, a sec θ cases

Why the topic needs it: every trig substitution ( $x = a sin θ$ , etc.) is really "let $x$ be one particular side of this triangle." Without the picture, the substitution is just a spell you memorise; with it, every step has a location you can point to.

2. Pythagoras — why squares appear at all

What it looks like: the area of the two smaller squares built on the short sides exactly fills the big square built on the hypotenuse (figure below).

Why the topic needs it: the three ugly radicals $a^{2} - x^{2}$ , $a^{2} + x^{2}$ , $x^{2} - a^{2}$ are Pythagoras rearranged to solve for the missing side. For example $a^{2} - x^{2}$ is "hypotenuse $a$ , one leg $x$ , find the other leg." That is why a triangle is always hiding inside these problems — see Reference right triangle method.

3. — the square-root symbol and its sign trap

The one rule that catches everybody: $u^{2} = ∣ u ∣ (the size of u, sign stripped off), NOT simply u .$

Why the topic needs it: in Step 2 of every derivation we write $a^{2} cos^{2} θ = a cos θ$ . That is only legal when $cos θ \geq 0$ , so that $∣ cos θ ∣ = cos θ$ . Guaranteeing that is the entire reason each substitution comes with a range of $θ$ (the last column of the section-0 table). We build those ranges carefully in Section 7 — for now just note: the absolute-value bars only vanish because $θ$ is confined to an interval where the trig quantity keeps one sign.

4. $a$ and $x$ — constant vs variable

How to find $a$ : it is the number sitting where $a^{2}$ appears. In $9 - x^{2}$ , $a^{2} = 9$ so $a = 3$ . In $x^{2} + 7$ , $a^{2} = 7$ so $a = 7$ .

Why the topic needs it: the substitution is always $x = a (trig)$ . Getting $a$ wrong scales every later step wrongly.

5. sin, cos, tan — three ratios ON the triangle

Now we attach the angle $θ$ to numbers. WHY these functions and not something else? Because for a fixed angle $θ$ , the ratios of the sides never change no matter how big the triangle is — so a ratio is a perfect fingerprint of the angle.

Why the topic needs it: $x = a sin θ$ literally means "let $x$ be the opposite side when the hypotenuse is $a$ ." $x = a tan θ$ means "let $x$ be opposite when adjacent is $a$ ." $x = a sec θ$ means "let $x$ be the hypotenuse when adjacent is $a$ ." Each substitution is a choice of which sides play which role.

6. The Pythagorean identities — the engines that kill the root

The three identities below are not new facts. Each one is Pythagoras, $sin^{2} θ + cos^{2} θ = 1$ , divided through by a different quantity. Here is the whole derivation, one line each. See Pythagorean identities for more.

Why the topic needs it — the whole trick in one line: a sum or difference of two squares under a root is hard, but a single square under a root is trivial ( $(stuff)^{2} = ∣ stuff ∣$ ). These identities collapse two terms into one squared term. That is the payoff, and it is why we pick trig substitutions rather than any random change of variable.

7. arcsin, arctan, arcsec — undoing the ratio, and where the ranges come from

Why a range is needed at all: $sin θ$ repeats forever, so "which angle" has infinitely many answers. We pin down ONE by restricting $θ$ . We choose the restriction so that (a) the domain of $x$ is fully covered and (b) the root stays $\geq 0$ . Here is each case with its reasoning:

$sin$ -case, $x = a sin θ$ . Since $- 1 \leq sin θ \leq 1$ , we get $- a \leq x \leq a$ , i.e. domain $∣ x ∣ \leq a$ . Choosing $θ \in [- \frac{π}{2}, \frac{π}{2}]$ sweeps $sin θ$ across all of $[- 1, 1]$ exactly once, and on that interval $cos θ \geq 0$ , so $a^{2} - x^{2} = a cos θ$ (no bars).
$tan$ -case, $x = a tan θ$ . As $θ$ runs over $(- \frac{π}{2}, \frac{π}{2})$ , $tan θ$ takes every real value once, so the domain is all real $x$ . On that interval $sec θ > 0$ , so $a^{2} + x^{2} = a sec θ$ (no bars).
$sec$ -case, $x = a sec θ$ . Since $∣ sec θ ∣ \geq 1$ , we get $∣ x ∣ \geq a$ , i.e. domain $∣ x ∣ \geq a$ . The standard choice is $θ \in [0, π] ∖ {\frac{π}{2}}$ : on $[0, \frac{π}{2})$ we cover $sec θ \geq 1$ (so $x \geq a$ ), and on $(\frac{π}{2}, π]$ we cover $sec θ \leq - 1$ (so $x \leq - a$ ). Caution: on $(\frac{π}{2}, π]$ we have $tan θ \leq 0$ , so $x^{2} - a^{2} = a ∣ tan θ ∣ = - a tan θ$ there. That sign flip (shown as $a ∣ tan θ ∣$ in the section-0 table) is exactly why the $sec$ -case needs extra care for negative $x$ .

Why the topic needs it: the final answer of every problem must be back in $x$ . Since $x = a sin θ$ gives $sin θ = \frac{x}{a}$ , the angle itself is $θ = arcsin \frac{x}{a}$ . Inverse trig is the only way to write $θ$ in terms of $x$ .

8. $\int f d x$ and $d x$ — the integral and its tail

Why the topic needs it: when we set $x = a sin θ$ , the width $d x$ must be rewritten in $θ$ . We do this by differentiating (Section 9). Forgetting the $d x$ conversion is the single most common trig-sub error.

9. The differential $d x = g^{'} (θ) d θ$ — how the width transforms

You only need three derivative facts, one per case (each is a standard rule, verified below): $x = a sin θ \Rightarrow d x = a cos θ d θ, x = a tan θ \Rightarrow d x = a sec^{2} θ d θ, x = a sec θ \Rightarrow d x = a sec θ tan θ d θ .$

Why the topic needs it: this is exactly the machinery of Integration by substitution (u-sub) — trig substitution is a u-sub where the new variable happens to be an angle. The transformed $d x$ usually cancels beautifully against the simplified root, which is the moment the integral becomes doable.

θ

Hypotenuse (opposite the right angle), opposite (across from

θ

), adjacent (touching

θ

State Pythagoras for a right triangle.

(opp)^{2} + (adj)^{2} = (hyp)^{2}

What does

u^{2}

equal — and why does the topic care?

∣ u ∣

; the

θ

-range makes the inside non-negative so the bars vanish.

x^{2} + 7

, what is

a

a = 7

(since

a^{2} = 7

Write

sin θ

cos θ

tan θ

as side ratios.

sin θ = opp / hyp

cos θ = adj / hyp

tan θ = opp / adj

What is

sec θ

in terms of

cos θ

sec θ = 1/ cos θ

Derive

tan^{2} θ + 1 = sec^{2} θ

from Pythagoras.

Divide

sin^{2} θ + cos^{2} θ = 1

cos^{2} θ

Give the domain of

x

for each substitution.

∣ x ∣ \leq a

for

a sin θ

; all real

x

for

a tan θ

;

∣ x ∣ \geq a

for

a sec θ

Give the standard range of

θ

for the

sec

-case.

θ \in [0, π] ∖ {\frac{π}{2}}

What question does

arcsin (r)

answer?

"Which angle

θ

has

sin θ = r

?" (within its range).

State the definition of the derivative

g^{'} (θ)

g^{'} (θ) = lim_{h \to 0} \frac{g ( θ + h ) - g ( θ )}{h}

, the instantaneous slope.

x = a tan θ

, what is

d x

d x = a sec^{2} θ d θ

Rewrite

cos^{2} θ

for integration.

cos^{2} θ = \frac{1 + cos 2 θ}{2}

Why is

d x

not optional in

\int f d x

It is the slice width; changing variable changes it, so it must be converted.

Connections

4.2.09 Trigonometric substitution — x = a sin θ, a tan θ, a sec θ cases (Hinglish) — parent topic, Hinglish version.
Pythagorean identities — the engine that kills the root.
Integration by substitution (u-sub) — trig sub is a u-sub in disguise.
Power-reduction & double-angle formulas — the finisher for $\int cos^{2} θ$ .
Reference right triangle method — how Section 1's picture powers back-substitution.
Hyperbolic substitution — the cousin method using $cosh, sinh$ .

Foundations — Trigonometric substitution — x = a sin θ, a tan θ, a sec θ cases