Trigonometry is a fundamental branch of mathematics that deals with relationships between angles and sides of triangles. For OTET Paper II, this topic focuses on trigonometric ratios of acute angles in right-angled triangles and the standard identities that connect these ratios. Questions typically test your ability to calculate ratio values, apply identities to simplify expressions, and solve problems involving heights and distances.
This topic carries significant weight in the Mathematics section and forms a bridge between geometry and algebra. Mastery of trigonometric ratios and identities is essential not only for direct questions but also for solving problems in mensuration and coordinate geometry. The syllabus expects you to know the six trigonometric ratios, their values at standard angles (0°, 30°, 45°, 60°, 90°), and the three fundamental identities.
Key Concepts
**Right-angled triangle terminology**: In a right triangle with angle θ, the side opposite to θ is the "opposite" (perpendicular), the side adjacent to θ is the "adjacent" (base), and the longest side facing the right angle is the "hypotenuse."
**Six trigonometric ratios**: For an acute angle θ in a right triangle:
sin θ = Opposite / Hypotenuse = P/H
cos θ = Adjacent / Hypotenuse = B/H
tan θ = Opposite / Adjacent = P/B
cosec θ = H/P (reciprocal of sin θ)
sec θ = H/B (reciprocal of cos θ)
cot θ = B/P (reciprocal of tan θ)
**Complementary angle relationship**: sin(90° - θ) = cos θ, cos(90° - θ) = sin θ, tan(90° - θ) = cot θ, and vice versa. This means ratios of complementary angles are related.
**Trigonometric identities are equalities**: They hold true for all values of the angle (within the domain) and are used to simplify complex expressions.
**Pythagorean connection**: The fundamental identities are derived from the Pythagorean theorem applied to a right triangle with hypotenuse 1.
**Ratio relationships**: tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. These conversion formulas help simplify mixed expressions.
Formulas / Key Facts
**Standard Trigonometric Ratios at Special Angles:**
Therefore: sin 55° - cos 35° = cos 35° - cos 35° = 0
Common Mistakes
**Confusing opposite and adjacent sides** → Always identify sides with respect to the specific angle θ, not the right angle. Draw the triangle and mark which angle you're working with.
**Using wrong reciprocal pairs** → Students often think sec is reciprocal of sin. Remember: **s**in pairs with co**s**ec (both have 's'), **c**os pairs with se**c** (both have 'c').
**Forgetting that tan 90° and cot 0° are undefined** → Division by zero occurs. When cos θ = 0, tan θ is undefined; when sin θ = 0, cot θ is undefined.
**Taking wrong sign of square root** → When finding cos θ from sin²θ + cos²θ = 1, remember that for acute angles (0° to 90°), all six ratios are positive. Only take the positive root.
**Misapplying complementary formulas** → sin(90° - θ) = cos θ, not sin θ. The function changes (sin ↔ cos, tan ↔ cot, sec ↔ cosec).
**Arithmetic errors with surds** → Rationalise denominators properly. 1/√3 = √3/3. Practice surd calculations separately.
Quick Reference
**SOH-CAH-TOA**: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent
**Identity 1**: sin²θ + cos²θ = 1 (the mother identity — others derive from this)
**sin 30° = cos 60° = 1/2** and **sin 60° = cos 30° = √3/2** (complementary pairs)
**At 45°, sin = cos = 1/√2** and tan = cot = 1 (the symmetric angle)