Trigonometry Cheat Sheet - CDS 2027 - Paper 3: Elementary Mathematics

Welcome to the comprehensive LibreTexts-style chapter guide for CDS 2027 Paper 3 (Elementary Mathematics) - Trigonometry. Trigonometry forms one of the most critical core components of the UPSC Combined Defence Services syllabus, consistently accounting for a significant fraction of total marks. This chapter provides an structured breakdown of fundamental ratios, trigonometric identities, quadrant rules, and standardized angle values needed to excel in CDS Paper 3.

1. Syllabus Analysis & Exam Weightage

In the UPSC CDS Elementary Mathematics paper, Trigonometry typically carries between 12 to 15 questions out of 100. Understanding both basic geometric definitions and quick conversion formulas allows candidates to solve these questions under strict time constraints. PYQ analysis reveals a heavy emphasis on algebraic simplification of trigonometric expressions, application of height and distance problems, and evaluating values in different quadrants.

2. Basic Definitions & Trigonometric Ratios

Trigonometric ratios are defined using a right-angled triangle with an angle \(\theta\), an Opposite side (\(O\)), an Adjacent side (\(A\)), and a Hypotenuse (\(H\)).

Primary Trigonometric Ratios

\[\sin\theta = \frac{O}{H}, \quad \cos\theta = \frac{A}{H}, \quad \tan\theta = \frac{O}{A}\]

Reciprocal Trigonometric Ratios

\[\csc\theta = \frac{H}{O}, \quad \sec\theta = \frac{H}{A}, \quad \cot\theta = \frac{A}{O}\]

Quotient Relations

\[\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}\]

3. Fundamental Pythagorean Identities

The core algebraic relations in trigonometry stem directly from the Pythagorean theorem (\(A^2 + O^2 = H^2\)). Mastering these identities helps quickly simplify complex terms during the examination.

• Primary Identity: \[\sin^2\theta + \cos^2\theta = 1\]
• Secant Identity: \[1 + \tan^2\theta = \sec^2\theta\]
• Cosecant Identity: \[1 + \cot^2\theta = \csc^2\theta\]

4. Unit Circle & ASTC Quadrant Rules

When extending angles beyond \(90^\circ\), the signs of trigonometric functions are governed by the quadrant in which the terminal side of the angle lies on the Cartesian plane.

The standard mnemonic to remember sign rules across the four quadrants is ASTC ("All Silver Tea Cups"):

• Quadrant I (0° to 90°): All trigonometric functions are positive.
• Quadrant II (90° to 180°): Sin (and its reciprocal \(\csc\)) are positive.
• Quadrant III (180° to 270°): Tan (and its reciprocal \(\cot\)) are positive.
• Quadrant IV (270° to 360°): Cos (and its reciprocal \(\sec\)) are positive.

5. Standard Angle Values

Memorizing key values for common angles (\(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\)) is necessary for rapid calculation in Paper 3.

Function	0°	30° (\(\frac{\pi}{6}\))	45° (\(\frac{\pi}{4}\))	60° (\(\frac{\pi}{3}\))	90° (\(\frac{\pi}{2}\))
\(\sin\theta\)	0	\(\frac{1}{2}\)	\(\frac{1}{\sqrt{2}}\)	\(\frac{\sqrt{3}}{2}\)	1
\(\cos\theta\)	1	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{\sqrt{2}}\)	\(\frac{1}{2}\)	0
\(\tan\theta\)	0	\(\frac{1}{\sqrt{3}}\)	1	\(\sqrt{3}\)	Undefined

6. Frequently Asked Questions (People Also Ask)