Reciprocal identities, quotient identities: Foundations

This page assumes you have seen nothing. Before you can trust a single line of the parent note, every mark on the page must mean something. Below is every symbol and idea the parent uses, built in order, each one a step on the previous.

1. The right triangle — the stage everything stands on

Figure — Reciprocal identities, quotient identities

Look at the figure. The little square in the bottom-left corner is the mark for a right angle — it says "these two sides meet like the edge of a table." We pick one of the other two corners to study and call its angle $θ$ (the Greek letter "theta", just a name for "the angle we care about").

Why do we need this shape? Because the moment one angle is fixed at $9 0^{\circ}$ , the three sides lock into fixed proportions for any given $θ$ . Those proportions are exactly the trig ratios. No right angle → no clean ratios.

2. The three sides: Opposite, Adjacent, Hypotenuse

Now that a corner $θ$ is chosen, the three sides get names relative to $θ$ — and this is the part beginners trip on, so we anchor each to the picture.

In the figure, stand at the corner $θ$ (red). Trace the two sides leaving that corner: one is the hypotenuse (the long slanted one), the other is the Adjacent — it lies right next to $θ$ . The side you are not touching, way across the triangle, is the Opposite.

Why these three names? Because the trig ratios are built from pairs of sides, and we need an unambiguous way to say "which side." "Opposite" and "adjacent" are defined from $θ$ 's point of view — move to the other corner and O and A swap. See Right triangle trigonometry (SOH-CAH-TOA).

3. What a fraction (ratio) actually says

The trig functions are all fractions like $\frac{O}{H}$ . Make sure this symbol is solid.

Why ratios and not raw lengths? A big triangle and a small triangle with the same angle $θ$ have different side lengths but the same ratios. The ratio throws away size and keeps only the shape — which is exactly the angle's steepness. That is the deep reason trig works at all.

4. The three fundamental functions: sin, cos, tan

The figure shows the same triangle three times, each highlighting the two sides that build one ratio (the key pair in red).

Why exactly these three pairings? There are three sides, so three natural pairs: (O,H), (A,H), (O,A). Those are precisely $sin, cos, tan$ . The mnemonic SOH-CAH-TOA stores them: Sine = Opp/Hyp, Cosine = Adj/Hyp, Tangent = Opp/Adj.

5. The reciprocal: flipping a fraction upside down

The parent's whole point is that three more functions come from flipping. So nail "flip."

Why does flipping matter here? Because $csc θ = \frac{H}{O}$ is literally $sin θ = \frac{O}{H}$ turned over. The whole "three extra functions" idea is just: flip the three you already have. No new triangle needed.

6. The superscript $- 1$ — a dangerous piece of notation

Why warn about this now? Because the parent lists a common mistake ( $csc θ = sin^{- 1} θ$ ) that comes entirely from this notation clash. The inverse function undoes sine (gives back an angle); the reciprocal just flips the ratio (gives another ratio). See Inverse trigonometric functions.

7. Square roots and $θ^{2}$ — for the Pythagorean helper

Example 2 in the parent uses $cos θ = 1 - sin^{2} θ$ . Two symbols hide there.

Why do we need these? The Pythagorean identities say $sin^{2} θ + cos^{2} θ = 1$ . Rearranged, $cos θ = 1 - sin^{2} θ$ recovers the missing ratio from the one you know. That is how Example 2 finds $cos$ from $sin$ alone. This identity itself comes from the Pythagoras rule $O^{2} + A^{2} = H^{2}$ on our triangle — and to place a point at $(3, 4)$ and read $H = 5$ , see Unit circle definitions.

8. Putting the flip and the divide together

You now hold every piece the parent uses:

A triangle with a chosen angle $θ$ (§1).
Three named sides O, A, H (§2).
Ratios that measure steepness (§3).
sin, cos, tan as the three fundamental ratios (§4).
Reciprocal = flip → gives csc, sec, cot (§5).
$- 1$ notation carefully distinguished (§6).
Square / square root for the Pythagorean helper (§7).

With these, the parent's "convert everything to sin and cos" trick becomes obvious: every function is either a ratio of two of {O, A, H} or a flip of one, so it can always be rewritten using just sin and cos.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer aloud.

In a right triangle, which side is always the hypotenuse?

The longest side, the one opposite the right angle.

The "opposite" and "adjacent" sides are named relative to what?

The chosen angle

θ

— swap corners and they swap.

Why do we use ratios of sides instead of the side lengths themselves?

Ratios ignore the triangle's size and capture only the angle's shape/steepness.

Write

sin θ

cos θ

tan θ

as side ratios.

sin θ = O / H

cos θ = A / H

tan θ = O / A

What does SOH-CAH-TOA stand for?

Sin=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj.

What does "reciprocal" do to

a / b

Flips it to

b / a

(i.e.

1

over it).

What is the difference between

(sin θ)^{- 1}

and

sin^{- 1} θ

(sin θ)^{- 1} = csc θ

is the reciprocal;

sin^{- 1} θ

is arcsine, the inverse function that returns an angle.

Evaluate

1 - \frac{4}{9}

5 /3

Is "sin" a multiplication of letters

s \cdot i \cdot n

No — it is one three-letter name for a function that eats an angle and returns a ratio.