Introduction to Trigonometry (Class 10)
About 7 min · Mathematical Reasoning · English
Time worked out from the lesson's 543 words at 150 words a minute, plus a minute for each question.
Trigonometry forms the bridge between geometry and algebra, dealing with relationships between sides and angles of triangles. For SOF IMO Class 10, this topic combines conceptual understanding with computational skill. Questions may test your grasp of the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent), complementary angle relationships, fundamental identities, and real-world applications through heights and distances problems.
Key ideas
- Trigonometric Ratios — In a right triangle with angle θ, the six ratios relate the sides: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent, and their reciprocals cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
- Domain Restrictions — Trigonometric ratios are defined only when denominators are non-zero; for example, tan 90° and sec 90° are undefined because cos 90° = 0.
- Complementary Angles — Two angles are complementary if they sum to 90°. Key relationships: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ, and vice versa for reciprocals.
- Pythagorean Identities — These stem from Pythagoras theorem applied to the unit circle: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ. These identities are essential for simplification and proof problems.
- Standard Angle Values — Memorize exact values for 0°, 30°, 45°, 60°, 90° for all six ratios. For instance, sin 30° = 1/2, cos 45° = 1/√2, tan 60° = √3, cosec 90° = 1.
Watch out for
- Confusing sin θ and cosec θ — Students often invert the wrong ratio. Remember: cosec θ is the reciprocal (1/sin θ), not the complementary angle formula. If sin 30° = 1/2, then cosec 30° = 2, not cos 30°.
- Wrong complementary angle substitution — Writing sin(90° − 30°) = sin 60° is correct, but some forget to simplify: sin(90° − 30°) = cos 30°. Always apply the complementary formula before plugging in values.
- Misremembering standard values — Mixing up sin 60° and cos 60° is common. Use mnemonic: sin increases from 0° to 90° (0 → 1), cos decreases (1 → 0). So sin 60° = √3/2 (larger) and cos 60° = 1/2 (smaller).
Quick reference
- Six Ratios: sin, cos, tan (primary); cosec, sec, cot (reciprocals). Know all definitions from a right triangle.
- Complementary Rule: sin(90° − θ) = cos θ and vice versa; extends to all six ratios.
- Three Pythagorean Identities: sin² + cos² = 1, 1 + tan² = sec², 1 + cot² = cosec². Memorize cold.
- Standard Angles: 0°, 30°, 45°, 60°, 90° — drill the table until instant recall.
- Heights/Distances Setup: Identify right triangle, label angle, pick correct ratio (usually tan for angle of elevation/depression).
Check yourself: 3 questions
1. If sin θ = 12/13, then what is the value of cos θ?
2. If tan(A + B) = √3 and tan(A - B) = 1/√3, where A and B are acute angles, then what is the value of A?
3. A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of the hill as 30°. What is the distance of the hill from the ship (in metres)?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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