Reciprocal identities — cosec, sec, cot in terms of sin, cos, tan: Foundations

Before you can understand reciprocal identities, you must own every symbol the parent note quietly assumes. We build them in order — each one uses only the ones before it.

0. The angle $θ$ (theta) — the star of the show

Picture two rays starting at the same corner. The gap between them is the angle. A small gap = a small $θ$ ; a wide gap = a large $θ$ .

Figure — Reciprocal identities — cosec, sec, cot in terms of sin, cos, tan

Figure s01 — Two black rays leave a shared vertex; the red arc between them is the angle $θ$ . Widen the top ray and the red arc grows; that growing arc IS the angle.

What $π$ means (so radians make sense)

We can measure the angle two ways:

Degrees ( $°$ ): a full turn is $360°$ , a square corner is $90°$ .
Radians: a full turn is $2 π$ (about $6.28$ ), a square corner is $\frac{π}{2}$ .

For now, just read $90° = \frac{π}{2}$ , $180° = π$ , $360° = 2 π$ . The parent note uses both — they name the same turn, in different units, like metres vs feet.

The allowed range of $θ$ for a triangle

1. The right triangle and its three sides

The whole topic lives inside one shape: a right triangle — a triangle with one square corner ( $90°$ ).

Figure s02 — A right triangle with the square corner at bottom-right. The angle $θ$ sits at bottom-left. The horizontal black side is the adjacent ( $b$ ), the vertical black side is the opposite ( $a$ ), and the red slanted side is the hypotenuse ( $r$ ). Notice the opposite side stands vertically and the adjacent side runs horizontally — remember these directions for "rise" and "run" below.

2. What a fraction (ratio) actually says

The parent note is made of fractions like $\frac{a}{r}$ . Make sure the fraction itself is solid.

Figure s03 — A small right triangle sits inside a larger copy with every side doubled. The shared angle $θ$ is marked at the common corner. The red labels show that $\frac{a}{r}$ in the small triangle equals $\frac{2 a}{2 r} = \frac{a}{r}$ in the big one — the size cancels, the ratio survives.

Look at the two nested triangles: the big one has every side doubled, yet $\frac{a}{r}$ is the same value in both. The red ratio label doesn't move. Angle in — same ratio out.

3. The three base functions: sine, cosine, tangent

Now the ratios earn names. These are borrowed from the definition of sine, cosine, tangent.

A note on signs (what happens beyond the triangle)

4. The reciprocal move: flipping a fraction

This is the single operation the parent topic is built on.

Worked flip: $\frac{1}{p / q} = 1 \times \frac{q}{p} = \frac{q}{p}$

Naming the three flipped functions

Applying the flip to $sin θ$ , $cos θ$ , $tan θ$ gives three brand-new functions — the very ones the parent topic is about. Define their symbols now, before using them anywhere.

So the flip move applied to $sin θ = \frac{a}{r}$ turns it into $\frac{r}{a}$ , and that flipped ratio is what we name $csc θ$ .

5. The exponent-superscript trap: $sin^{2} θ$ vs $sin^{- 1} θ$

The parent note writes $sin^{2} θ$ (in Pythagoras) and warns about $sin^{- 1} θ$ . Two different meanings hide behind the small raised number.

6. The Pythagorean identity (a tool the parent borrows)

Example 3 and Example 4 in the parent lean on this. It comes from the Pythagorean identities.

We need it because knowing $cos θ$ alone lets us recover $sin θ$ — and therefore $csc θ$ — without re-measuring the triangle.

How these feed the topic

Read the map below top to bottom: each box is a foundation from this page, and the arrows show what it feeds into. Start at "Angle theta" (top-left) and follow the arrows — you literally cannot reach "Reciprocal identities" at the bottom without passing through every box above it. Use it as a checklist: if any box feels shaky, jump back to its section.

The map shows two streams meeting: the triangle stream (angle → triangle → sides → ratios) and the algebra stream (fractions → the flip). They join to create the reciprocal functions the parent note names.

Equipment checklist

Cover the right side and test yourself. If any line surprises you, re-read its section above.

What does the symbol

θ

stand for?

A name for an angle — the amount of turn between two rays.

What is

π

A fixed number (about

3.14159

) equal to a circle's circumference divided by its diameter.

What range of

θ

does this page's triangle allow?

Acute only:

0° < θ < 90°

(that is

0 < θ < \frac{π}{2}

Which side is the hypotenuse defined by?

The square (

90°

) corner — it is always opposite the right angle, never chosen by

θ

What does "opposite" mean relative to

θ

The side that does not touch

θ

, sitting across from it.

Why must the bottom

q

of a ratio

\frac{p}{q}

be non-zero?

A zero bottom,

\frac{p}{0}

, has no value; the fraction is undefined.

Why do we use ratios of sides instead of raw lengths?

Ratios ignore the triangle's size and capture only its shape (the angle); similar triangles share identical ratios.

Write

sin θ

cos θ

tan θ

as side ratios.

sin θ = \frac{opp}{hyp}

cos θ = \frac{adj}{hyp}

tan θ = \frac{opp}{adj}

In the triangle, which side is the "rise" and which is the "run"?

Rise = vertical opposite side

a

; run = horizontal adjacent side

b

;

tan θ = \frac{rise}{run}

Define

csc θ

sec θ

cot θ

as flipped ratios.

csc θ = \frac{1}{s i n θ} = \frac{r}{a}

sec θ = \frac{1}{c o s θ} = \frac{r}{b}

cot θ = \frac{1}{t a n θ} = \frac{b}{a}

What is the reciprocal of

\frac{p}{q}

\frac{q}{p}

— flip it upside-down (needs

p \neq = 0

Why is

\frac{1}{0}

undefined rather than infinity?

No number of copies of

0

ever adds up to

1

; the value simply does not exist.

In which quadrant are all of

sin

cos

tan

positive?

Quadrant I (

0°

–

90°

) — our acute triangle range.

What does

sin^{2} θ

mean?

(sin θ)^{2}

— take the sine, then square that number.

sin^{- 1} θ

the same as

\frac{1}{s i n θ}

No.

sin^{- 1} θ = arcsin θ

is the inverse function (gives an angle); the reciprocal

\frac{1}{s i n θ}

csc θ

State the Pythagorean identity and where it comes from.

sin^{2} θ + cos^{2} θ = 1

, obtained by dividing

a^{2} + b^{2} = r^{2}

r^{2}

When every line is second nature, you are ready for the reciprocal identities themselves, and later for using them inside basic trig equations and derivatives of trig functions.