Trigonometric ratios in right triangle — sin, cos, tan, cosec, sec, cot: Foundations

Every symbol you will later read — the ones spelled $sin$ , $cos$ , $tan$ , the Greek letter $θ$ , and the side-names "opposite", "adjacent", "hypotenuse" — is defined further down this very page. We build them from the ground up, so read top to bottom and nothing will be a surprise.

0. The very first picture: what "triangle" and "angle" even mean

A triangle is a closed shape with three straight sides and three corners. A corner is called a vertex (plural: vertices).

At each corner, the two sides that meet there open up like a pair of scissors. How wide they open is the angle at that corner. We measure that opening in degrees, written with a little circle: $3 0^{\circ}$ , $9 0^{\circ}$ , and so on. A full turn all the way around is $36 0^{\circ}$ ; a half turn is $18 0^{\circ}$ ; a quarter turn — a perfect "square corner" — is $9 0^{\circ}$ .

Figure — Trigonometric ratios in right triangle — sin, cos, tan, cosec, sec, cot

Figure s01 — A horizontal side and a slanted side meet at a corner; the orange arc marks how wide they open (the angle). On the right, two sides meet in a perfect square corner, marked by a red square, labelled "right angle = 90 degrees".

WHY we need this: every ratio on this page is defined only for right triangles. The right angle is the anchor that gives the other two sides fixed roles. No right angle → the side-names below do not apply.

1. The symbol $θ$ (theta)

$θ$ is a Greek letter, said "THAY-ta". It is not a number you are given — it is a name we hand to one particular angle so we can talk about it. Think of it like calling a person "the tall one": it points at a specific thing without measuring it yet.

Intuition

$θ$ must be acute — strictly between $0^{\circ}$ and $9 0^{\circ}$ A triangle's three angles always add up to $18 0^{\circ}$ . One of them is already the right angle, using up $9 0^{\circ}$ , so the two remaining angles must share the leftover $9 0^{\circ}$ between them. That forces each of them to be more than $0^{\circ}$ and less than $9 0^{\circ}$ . An angle in that range ( $0^{\circ} < θ < 9 0^{\circ}$ ) is called acute. So the corner you name $θ$ is always acute.

The two forbidden edge cases (and why the right triangle breaks):

If $θ = 0^{\circ}$ , the "triangle" has flattened into a straight line — the corner has no opening, and there is no genuine side "across from $θ$ ". No real triangle exists.
If $θ = 9 0^{\circ}$ , then $θ$ is the right angle, leaving $0^{\circ}$ for the last corner — again a flattened line, not a triangle.

So the right-triangle definitions only make sense for a genuinely acute $θ$ ; the values $0^{\circ}$ and $9 0^{\circ}$ are handled later by a wider definition, not by triangle sides.

WHY: all the ratios below say "for angle $θ$ ". If you don't know $θ$ is a chosen acute label (not a fixed value, and never $0^{\circ}$ or $9 0^{\circ}$ ), the phrase "opposite side" has no meaning — opposite to what?

2. The three sides get names — but only after $θ$ is chosen

Once $θ$ is picked, the three sides split into three roles. Look at the figure and read the colours.

Figure s02 — A right triangle with the square corner at the bottom-right (red square). The chosen acute angle $θ$ sits at the bottom-left. The bottom side (green) touches $θ$ and is labelled "adjacent". The vertical side (orange) is across from $θ$ and labelled "opposite". The long slanted side (blue) is across from the square corner and labelled "hypotenuse".

WHY we need this: the six ratios below are built entirely from these three roles. If you mislabel a side, every single ratio comes out wrong.

3. The fraction bar and the word "ratio"

A ratio is a comparison of two amounts by division: "how many times does the bottom fit into the top?" We write it as one number stacked on another with a bar between:

$\frac{top}{bottom}$

read "top over bottom". If the top is $3$ and the bottom is $4$ , the fraction $\frac{3}{4} = 0.75$ says the top is three-quarters the size of the bottom.

WHY: every trig ratio is a fraction of two side-lengths. The whole subject is fractions of sides. If the fraction bar is fuzzy, nothing downstream is clear.

4. Why a ratio of sides depends only on the angle — the Similar triangles idea

Here is the deepest foundation of the whole topic. Blow a right triangle up to double size, like a photocopy at 200%. Every side doubles. The corners — the angles — stay exactly the same.

Figure s03 — Two right triangles side by side sharing the same acute angle $θ$ : a small blue one (opposite $= 1.5$ , hypotenuse $= 2.5$ ) and a big orange one at double scale (opposite $= 3$ , hypotenuse $= 5$ ). Text shows both give opposite $\div$ hypotenuse $= 0.6$ , so the same angle gives the same ratio.

Two triangles with the same angles are called similar — same shape, different size. In similar triangles, matching sides grow by the same factor $k$ . So take any two sides and form their ratio:

$\frac{opposite}{hypotenuse} = \frac{k \cdot ( opposite )}{k \cdot ( hypotenuse )} = \frac{opposite}{hypotenuse}$

The $k$ on top and the $k$ on bottom cancel. WHAT we just did: scaled the triangle and watched the ratio survive untouched. WHY it matters: it proves the ratio is glued to the angle, not the size. WHAT it looks like: the small and big triangles in the figure give the identical fraction.

5. The names $sin, cos, tan$ and their reciprocals

Now the ratios get short written names. These are just labels for specific side-fractions — no new maths, only new spelling.

The other three are reciprocals — a reciprocal just means "flip the fraction upside down" ( $\frac{3}{4} \to \frac{4}{3}$ ).

WHY six and not three: three of them are just upside-down versions kept around because they make some later formulas cleaner. Everything you need is $sin, cos, tan$ .

6. The symbol $=$ , the symbol $⟹$ , squaring $a^{2}$ , and the square root

Four tiny bits of grammar you will lean on:

WHY: Pythagoras theorem, written $a^{2} + b^{2} = c^{2}$ , uses $=$ , the exponent, and $⟹$ all at once — and to actually get the side length $c$ out of $c^{2}$ we must undo the square with . So all four are needed before the next section.

7. Pythagoras theorem — the tool that finds the missing side

Real problems often give you two sides of a right triangle and expect you to find the third. Here is the tool that does it, from zero.

Figure s04 — A 3-4-5 right triangle with a real square drawn outward on each side: a green $3 \times 3$ square (area 9) on one leg, a blue $4 \times 4$ square (area 16) on the other leg, and an orange $5 \times 5$ square (area 25) on the hypotenuse. Text notes $9 + 16 = 25$ , so the two small square areas together fill the big one.

WHY we need it: the ratios need all three sides. Real problems give you two; Pythagoras hands you the third. Example: legs $3$ and $4$ give $c^{2} = 3^{2} + 4^{2} = 25$ , and undoing the square with the root gives $c = 25 = 5$ .

8. Putting it together — the prerequisite map

The diagram below is a flow map: read it top to bottom. Each box is one idea from this page, and an arrow ( $\to$ ) means "the idea at the tail is needed before the idea at the head". So follow the arrows to see which foundations must be in place before the six ratios can exist. If your reader sees only code and no diagram, that means their app has not rendered the Mermaid block — the intended reading is the pictured flow, degrees → right angle → right triangle, joining with the chosen $θ$ and side-roles, then similar triangles and Pythagoras, all pouring into the six ratios.

In words: degrees give the right angle; the right angle plus a chosen acute $θ$ give the side-roles; similar triangles make the ratios size-free; and Pythagoras supplies any missing length — all feeding into the six ratios.

A tiny worked check (using only this page)

9. All six ratios in one place

Name	Spoken	Ratio (side form)	Reciprocal of
$sin θ$	"sine"	$o / h$	—
$cos θ$	"cosine"	$a / h$	—
$tan θ$	"tangent"	$o / a$	—
$csc θ$	"cosecant"	$h / o$	$sin θ$
$sec θ$	"secant"	$h / a$	$cos θ$
$cot θ$	"cotangent"	$a / o$	$tan θ$

Read the table as three primary ratios (top three rows) and their three upside-down partners (bottom three rows). This is the full toolkit that the parent topic builds everything else on.

Equipment checklist

Cover the right side and answer out loud; reveal to check.

What does the small square drawn in a triangle's corner mean?

That corner is exactly a

9 0^{\circ}

right angle.

What is

θ

in a right triangle, and what range must it lie in?

A name we give to one chosen non-right corner; it must be acute, strictly between

0^{\circ}

and

9 0^{\circ}

Why can

θ

never be

0^{\circ}

9 0^{\circ}

in a right triangle?

The three angles total

18 0^{\circ}

; the right angle takes

9 0^{\circ}

, so the other two must split the remaining

9 0^{\circ}

— at those extremes the triangle flattens to a line.

Which side is the hypotenuse, and does it depend on

θ

The side across from the right angle (always the longest); it never changes when

θ

moves.

Which leg is "opposite" and which is "adjacent"?

Opposite = the leg not touching

θ

; adjacent = the leg touching

θ

that is not the hypotenuse.

Why does a side-ratio depend only on the angle, not the triangle's size?

Because scaling by

k

multiplies top and bottom equally and cancels — similar triangles share ratios.

What does the fraction bar

\frac{A}{B}

tell you?

How big

A

is compared to

B

— the single number you get by dividing

A

B

Write

sin θ

cos θ

tan θ

as side-ratios.

sin θ = \frac{o}{h}

cos θ = \frac{a}{h}

tan θ = \frac{o}{a}

What are

csc, sec, cot

The flipped (reciprocal) versions of

sin, cos, tan

\frac{h}{o}, \frac{h}{a}, \frac{a}{o}

sin θ

a multiplication of

sin

and

θ

No — it is one indivisible symbol;

sin

alone means nothing.

What does

25 = 5

mean?

The positive number which, multiplied by itself, gives

25

— the square root undoes squaring.

State Pythagoras' theorem and give its area picture-proof in one line.

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

; the square built on the hypotenuse has area equal to the two squares on the legs added together.