Trigonometry

Overview

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:**

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

**Memory tip for sin values**: √0/2, √1/2, √2/2, √3/2, √4/2 gives 0, 1/2, 1/√2, √3/2, 1 for 0°, 30°, 45°, 60°, 90°.