📐 Introduction to Trigonometry — Class 10

Trigonometric ratios, values of specific angles, identities, and complementary angles

1. Trigonometric Ratios

Trigonometry deals with the relationship between the angles and sides of a right-angled triangle. For an acute angle θ in a right triangle:

📖 The Six Trigonometric Ratios

In right triangle ABC with right angle at B and angle θ at A:

Perpendicular (P) = side opposite to angle θ (BC)

Base (B) = side adjacent to angle θ (AB)

Hypotenuse (H) = side opposite to right angle (AC)

• sin θ = P/H = BC/AC

• cos θ = B/H = AB/AC

• tan θ = P/B = BC/AB

• cosec θ = H/P = 1/sin θ

• sec θ = H/B = 1/cos θ

• cot θ = B/P = 1/tan θ

🌟 Memory Trick — SOH CAH TOA

SOH: Sin = Opposite/Hypotenuse
CAH: Cos = Adjacent/Hypotenuse
TOA: Tan = Opposite/Adjacent
Say it fast: "So-Kah-Toe-Ah" and you'll never forget!

💡 Example: Find all trig ratios if sin θ = 3/5

P = 3, H = 5 → B = √(5² – 3²) = √(25–9) = √16 = 4

sin θ = 3/5 | cos θ = 4/5 | tan θ = 3/4

cosec θ = 5/3 | sec θ = 5/4 | cot θ = 4/3

2. Values of Trigonometric Ratios for Specific Angles

Ratio	0°	30°	45°	60°	90°
sin	0	1/2	1/√2	√3/2	1
cos	1	√3/2	1/√2	1/2	0
tan	0	1/√3	1	√3	undefined
cosec	undefined	2	√2	2/√3	1
sec	1	2/√3	√2	2	undefined
cot	undefined	√3	1	1/√3	0

🔑 Memory Trick for sin values: 0°, 30°, 45°, 60°, 90°

sin values: √0/2, √1/2, √2/2, √3/2, √4/2 → 0, 1/2, 1/√2, √3/2, 1
cos values: opposite order of sin values
tan = sin/cos

💡 Example: Evaluate 2sin²30° + cos²45° – 4tan²30°

= 2×(1/2)² + (1/√2)² – 4×(1/√3)²

= 2×1/4 + 1/2 – 4×1/3

= 1/2 + 1/2 – 4/3

= 1 – 4/3 = –1/3

3. Trigonometric Identities

📖 The Three Fundamental Identities

Identity 1: sin²θ + cos²θ = 1

→ sin²θ = 1 – cos²θ and cos²θ = 1 – sin²θ

Identity 2: 1 + tan²θ = sec²θ

→ sec²θ – tan²θ = 1 and tan²θ = sec²θ – 1

Identity 3: 1 + cot²θ = cosec²θ

→ cosec²θ – cot²θ = 1 and cot²θ = cosec²θ – 1

🌟 Derivation of Identity 1

In right triangle: P² + B² = H² (Pythagoras theorem)

Dividing both sides by H²: P²/H² + B²/H² = 1

→ sin²θ + cos²θ = 1 ✓ (Simple and elegant!)

💡 Example: Prove (sinθ + cosθ)² + (sinθ – cosθ)² = 2

LHS = sin²θ + 2sinθcosθ + cos²θ + sin²θ – 2sinθcosθ + cos²θ

= (sin²θ + cos²θ) + (sin²θ + cos²θ)

= 1 + 1 = 2 = RHS ✓

💡 Example: Prove cosA/(1–tanA) + sinA/(1–cotA) = sinA + cosA

LHS = cosA/(1–sinA/cosA) + sinA/(1–cosA/sinA)

= cosA/[(cosA–sinA)/cosA] + sinA/[(sinA–cosA)/sinA]

= cos²A/(cosA–sinA) + sin²A/(sinA–cosA)

= cos²A/(cosA–sinA) – sin²A/(cosA–sinA)

= (cos²A – sin²A)/(cosA–sinA) = (cosA+sinA)(cosA–sinA)/(cosA–sinA)

= sinA + cosA = RHS ✓

4. Trigonometric Ratios of Complementary Angles

📖 Complementary Angle Identities

Two angles are complementary if their sum is 90°. For an angle θ:

• sin(90°–θ) = cosθ | cos(90°–θ) = sinθ

• tan(90°–θ) = cotθ | cot(90°–θ) = tanθ

• sec(90°–θ) = cosecθ | cosec(90°–θ) = secθ

Complementary functions are: sin↔cos, tan↔cot, sec↔cosec

💡 Example: Evaluate without tables: sin68°/cos22° + cos57°/sin33°

sin68° = sin(90°–22°) = cos22°

cos57° = cos(90°–33°) = sin33°

= cos22°/cos22° + sin33°/sin33° = 1 + 1 = 2

📋 Trigonometry Formula Sheet

Ratios

sin = P/H, cos = B/H

tan = P/B = sin/cos

Identities

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = cosec²θ

Complementary

sin(90–θ) = cosθ

tan(90–θ) = cotθ

sec(90–θ) = cosecθ

Special Values

sin30=1/2, cos30=√3/2

sin45=1/√2, cos45=1/√2

sin60=√3/2, cos60=1/2