Introduction to Trigonometry
Trigonometric ratios, standard angles, and complementary angle relations.
Chapter Overview
Remember SOH-CAH-TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
Trigonometry studies relationships between angles and sides of right triangles. Class 10 introduces sin, cos, tan ratios, standard values at 0°, 30°, 45°, 60°, 90°, and complementary angle identities.
Trigonometric Ratios
sin θ
opposite / hypotenuse
cos θ
adjacent / hypotenuse
tan θ
opposite / adjacent = sin θ / cos θ
Reciprocals
cosec = 1/sin, sec = 1/cos, cot = 1/tan
Standard Values
- sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3
- sin 45° = cos 45° = 1/√2, tan 45° = 1
- sin 60° = √3/2, cos 60° = ½, tan 60° = √3
- sin 90° = 1, cos 90° = 0
Complementary Angles
- sin(90° − θ) = cos θ
- cos(90° − θ) = sin θ
- tan(90° − θ) = cot θ
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Find ratios
In right ΔABC, ∠B = 90°, AB = 3, BC = 4. Find sin C and cos C.
- 1AC = √(3² + 4²) = 5
- 2sin C = opposite/hyp = AB/AC = 3/5
- 3cos C = adjacent/hyp = BC/AC = 4/5
Answer
sin C = 3/5, cos C = 4/5
Example 2: Standard value
Evaluate sin²45° + cos²45°.
- 1sin 45° = cos 45° = 1/√2
- 2sin²45° + cos²45° = ½ + ½
Answer
1
Key Points to Remember
- ✓ Ratios defined only for acute angles in right triangle (Class 10)
- ✓ sin²θ + cos²θ = 1 (Pythagorean identity)
- ✓ tan θ undefined when cos θ = 0 (θ = 90°)
Exam Tips
- • Draw right triangle and label opposite/adjacent relative to angle
- • Memorise standard value table
- • Rationalise denominators when needed (e.g. 1/√3)
Formula Cheat Sheet
Quick reference — NCERT board exam formulas
Reciprocal & Quotient
- sin θ = 1/cosec θ ; cosec θ = 1/sin θ
- cos θ = 1/sec θ ; sec θ = 1/cos θ
- tan θ = 1/cot θ ; cot θ = 1/tan θ
- tan θ = sin θ/cos θ ; cot θ = cos θ/sin θ
Pythagorean Identities
- sin²θ + cos²θ = 1
- tan²θ + 1 = sec²θ
- cot²θ + 1 = cosec²θ
Angle Addition
- sin(A ± B) = sin A cos B ± cos A sin B
- cos(A ± B) = cos A cos B ∓ sin A sin B
- tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
Double & Triple Angle
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
- tan 2θ = 2 tan θ/(1 − tan²θ)
- sin 3θ = 3 sin θ − 4 sin³θ
- cos 3θ = 4 cos³θ − 3 cos θ
Power Reducing (Integrals)
- sin²θ = (1 − cos 2θ)/2
- cos²θ = (1 + cos 2θ)/2
- tan²θ = (1 − cos 2θ)/(1 + cos 2θ)
Product ↔ Sum
- sin A sin B = ½[cos(A−B) − cos(A+B)]
- cos A cos B = ½[cos(A−B) + cos(A+B)]
- sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2)
- cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2)
Half Angle & Triangle Laws
- sin(θ/2) = ±√[(1 − cos θ)/2]
- Arc length: S = rθ (θ in radians)
- a/sin A = b/sin B = c/sin C (sine rule)
- a² = b² + c² − 2bc cos A (cosine rule)
Practice MCQs — Introduction to Trigonometry
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
sin 90° equals: