Cheat Sheet — Introduction to Trigonometry

📐 Trigonometric Ratios

IN RIGHT △ABC, ∠B = 90°, FOR ANGLE A (= θ)

sin θ = O/H cos θ = A/H tan θ = O/A

O = BC (opposite), A = AB (adjacent), H = AC (hypotenuse)

RECIPROCAL RATIOS

cosec θ = H/O sec θ = H/A cot θ = A/O

sin·cosec = 1 cos·sec = 1 tan·cot = 1

tan θ = sin θ / cos θ cot θ = cos θ / sin θ

Mnemonic: SOH-CAH-TOA or "Some People Have Curly Brown Hair Through Proper Brushing"

Hypotenuse is always opposite the 90° angle — it never changes.

📊 Standard Angle Values Table

θ	0°	30°	45°	60°	90°
sin θ	0	1/2	1/√2	√3/2	1
cos θ	1	√3/2	1/√2	1/2	0
tan θ	0	1/√3	1	√3	N.D.
cosec θ	N.D.	2	√2	2/√3	1
sec θ	1	2/√3	√2	2	N.D.
cot θ	N.D.	√3	1	1/√3	0

Memory Trick for sin sin 0°, 30°, 45°, 60°, 90° = √0/2, √1/2, √2/2, √3/2, √4/2. cos = same in reverse order. tan = sin ÷ cos.

🔺 Special Triangles

30-60-90 TRIANGLE (half of equilateral △, side 2)

Sides: 1, √3, 2

sin 30°=1/2, cos 30°=√3/2, tan 30°=1/√3
sin 60°=√3/2, cos 60°=1/2, tan 60°=√3

45-45-90 TRIANGLE (isosceles right triangle)

Sides: 1, 1, √2

sin 45° = cos 45° = 1/√2, tan 45° = 1

These two triangles give ALL standard angle values!

🔗 Trigonometric Identities

IDENTITY ① (MASTER IDENTITY)

sin²θ + cos²θ = 1

Proof: Pythagoras → O²+A²=H² → divide by H²

IDENTITY ② (divide ① by cos²θ)

1 + tan²θ = sec²θ

IDENTITY ③ (divide ① by sin²θ)

1 + cot²θ = cosec²θ

Useful: sec²θ − tan²θ = 1 → (secθ−tanθ)(secθ+tanθ) = 1

Useful: cosec²θ − cot²θ = 1 → (cosecθ−cotθ)(cosecθ+cotθ) = 1

🔄 Complementary Angles (A + B = 90°)

COMPLEMENTARY ANGLE FORMULAS

sin(90°−θ) = cos θ cos(90°−θ) = sin θ

tan(90°−θ) = cot θ sec(90°−θ) = cosec θ

Each ratio → its "co-" partner. The "co" in cosine = complement's sine.

sin 72° = cos 18° | tan 55° = cot 35° | sec 80° = cosec 10°

Key result: sin²A + sin²(90°−A) = sin²A + cos²A = 1

📏 Range of Trig Ratios (0° ≤ θ ≤ 90°)

Ratio	Range	Notes
sin θ	0 to 1	Increases: 0→1
cos θ	1 to 0	Decreases: 1→0
tan θ	0 to ∞	Increases: 0→N.D.
cosec θ	∞ to 1	cosec θ ≥ 1
sec θ	1 to ∞	sec θ ≥ 1
cot θ	∞ to 0	Decreases

Quick Check If your answer gives sin θ = 2 or cos θ = −3 for an acute angle → it's WRONG.

✏️ Strategy for Proving Identities

Start from the more complex side (usually LHS)

Convert everything to sin θ and cos θ

Use sin²θ + cos²θ = 1 to simplify

Factor, combine fractions, rationalise denominators

Never cross the = sign — don't assume the result!

Conjugate trick: Multiply by (secθ+tanθ)/(secθ+tanθ) or (1+sinθ)/(1+sinθ) to simplify.

📝 Worked Examples

Given tan θ = 3/4, find all ratios:
O=3, A=4, H=√(9+16)=5
sin=3/5, cos=4/5, cosec=5/3, sec=5/4, cot=4/3

Evaluate sin 65°/cos 25°:
sin 65° = cos 25° → answer = 1

Prove (1+tan²θ)/(1+cot²θ) = tan²θ:
= sec²θ/cosec²θ = sin²θ/cos²θ = tan²θ ■

sin(A+B)=1, cos(A−B)=√3/2 → find A, B:
A+B=90°, A−B=30° → A=60°, B=30°

⚠️ Common Mistakes to Avoid

sin²θ ≠ sin(θ²) — sin²θ means (sin θ)² i.e. square of sine value

sin(A+B) ≠ sin A + sin B — trig functions are NOT linear!

√(sin²θ + cos²θ) ≠ sin θ + cos θ — √1 = 1, not sin θ + cos θ

tan 90° is undefined, not infinity — division by zero is not defined

1/sin θ = cosec θ, NOT sin⁻¹θ (that's inverse sine — different concept)

🔢 Common Pythagorean Triplets

O	A	H	If tan θ = O/A
3	4	5	sin=3/5, cos=4/5
5	12	13	sin=5/13, cos=12/13
8	15	17	sin=8/17, cos=15/17
7	24	25	sin=7/25, cos=24/25

Multiples also work: 6-8-10, 9-12-15, 10-24-26, etc.

🧠 Memory Tricks & Mnemonics

sin Table: "√0, √1, √2, √3, √4 — all over 2"

sin 0°=√0/2=0, sin 30°=√1/2=1/2, sin 45°=√2/2=1/√2, sin 60°=√3/2, sin 90°=√4/2=1

cos = sin in Reverse

cos 0°=1, cos 30°=√3/2, cos 45°=1/√2, cos 60°=1/2, cos 90°=0 — exactly reversed!

"co-" means "complement"

co-sine = complement's sine. co-tangent = complement's tangent. co-secant = complement's secant.

Indian Mnemonic: "Pandit Badri Prasad Har Har Bole"

sin=P/H, cos=B/H, tan=P/B (P=perpendicular, B=base, H=hypotenuse)

Identity Derivation: "Divide by cos² → Identity ②, Divide by sin² → Identity ③"

Both come from sin²θ+cos²θ=1. One master identity gives you all three!

As θ increases (0°→90°): sin ↑, cos ↓, tan ↑

Use this to quick-check your answers. If sin 60° < sin 30° in your answer, something is wrong.