Trigonometric
Ratios

📐
Concept
sin, cos & tan
Three ratios, sides explained, SOHCAHTOA.
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📏
Theorem
Pythagoras
h²=p²+b², find each side with the formula.
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✏️
Prove
Prove Identities
Step-by-step proofs: sin²θ+cos²θ=1 and more.
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🔺
Figure
Figure Questions
Given a triangle figure, find trig ratios step by step.
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∠B = 90° (bottom-left) · θ at C (bottom-right)
In right-angled △ABC, ∠B = 90°

∠ACB = θ is the reference angle at C. Each side is named by its position relative to θ.

h — AC
Hypotenuse (h)Longest side, opposite the right angle at B. Never touches the 90° corner.
p — AB
Perpendicular (p)Opposite the reference angle θ. Stands vertically.
b — BC
Base (b)Adjacent (next to) the reference angle θ. Lies flat at the bottom.
The Three Trigonometric Ratios
sin
Sine — sin θ
Opposite ÷ Hypotenuse
Sides
ABAC
Letters
ph
Words
perpendicularhypotenuse
= opp / hyp
cos
Cosine — cos θ
Adjacent ÷ Hypotenuse
Sides
BCAC
Letters
bh
Words
basehypotenuse
= adj / hyp
tan
Tangent — tan θ
Opposite ÷ Adjacent
Sides
ABBC
Letters
pb
Words
perpendicularbase
= opp / adj
Reciprocal Ratios
cosec θ
= 1sin θ = hp = hypopp
sec θ
= 1cos θ = hb = hypadj
cot θ
= 1tan θ = bp = adjopp
Memory Trick — SOHCAHTOA
SOH
Sin=Opp/Hyp
CAH
Cos=Adj/Hyp
TOA
Tan=Opp/Adj
Sin=Opp/Hyp  |  Cos=Adj/Hyp  |  Tan=Opp/Adj
Right angle at B · θ at C
h2 = p2 + b2
hypotenuse² = perpendicular² + base²
According to Pythagoras' Theorem:
h² = p² + b²
(Hypotenuse)² = (Perp)² + (Base)²
Therefore, each side:
h
Hypotenuse (h)
h = √(p² + b²)
Square root of (p² + b²)
p
Perpendicular (p)
p = √(h² − b²)
From h²=p²+b² → p²=h²−b²
b
Base (b)
b = √(h² − p²)
From h²=p²+b² → b²=h²−p²
Example: p=3, b=4 → h=√(9+16)=√25=5
p=12, h=13 → b=√(169−144)=√25=5
h = √(p²+b²)
Longest · opposite 90°
p = √(h²−b²)
Vertical · opposite θ
b = √(h²−p²)
Horizontal · adjacent θ