Trigonometry Cheat Sheet - CDS 2027

Welcome to the ultimate chapter-wise breakdown and study material for CDS 2027 Trigonometry. This guide maps out all fundamental identities, special angle values, compound angle transformations, and non-right triangle laws required to crack the Combined Defence Services examination.

1. Exam Pattern & PYQ Analysis

In the UPSC CDS (Combined Defence Services) Mathematics paper, Trigonometry constitutes approximately 12% to 15% of the total questions. A comprehensive analysis of previous years' question papers (PYQs) reveals a consistent emphasis on simplifying algebraic expressions involving trigonometric functions, applying conditional identities, and solving non-right triangle geometry problems.

Chapter Weightage & Trends

• Direct Application of Identities: 4–6 Questions
• Heights and Distances & Triangle Laws: 3–4 Questions
• Value-Based Expressions (Special Angles): 2–3 Questions

2. Right Triangle Trigonometric Definitions

Consider a right-angled triangle with acute angle \(\theta\), opposite side \(O\), adjacent side \(A\), and hypotenuse \(H\). The six primary trigonometric ratios are defined as follows:

\[\sin(\theta) = \frac{O}{H}, \quad \csc(\theta) = \frac{H}{O}\] \[\cos(\theta) = \frac{A}{H}, \quad \sec(\theta) = \frac{H}{A}\] \[\tan(\theta) = \frac{O}{A}, \quad \cot(\theta) = \frac{A}{O}\]

3. Standard Angle Values

Memorizing the exact values of trigonometric functions at key benchmark angles is critical for fast calculation during the CDS exam environment.

\(\theta\) (Deg)	\(\theta\) (Rad)	\(\sin\)	\(\cos\)	\(\tan\)
\(0^\circ\)	\(0\)	\(0\)	\(1\)	\(0\)
\(30^\circ\)	\(\frac{\pi}{6}\)	\(\frac{1}{2}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{\sqrt{3}}\)
\(45^\circ\)	\(\frac{\pi}{4}\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{\sqrt{2}}{2}\)	\(1\)
\(60^\circ\)	\(\frac{\pi}{3}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{2}\)	\(\sqrt{3}\)
\(90^\circ\)	\(\frac{\pi}{2}\)	\(1\)	\(0\)	Undefined

4. Fundamental Pythagorean Identities

These core algebraic equivalences form the basis for simplifying trigonometric equations in CDS question papers:

\[\sin^2(\theta) + \cos^2(\theta) = 1\] \[1 + \tan^2(\theta) = \sec^2(\theta)\] \[1 + \cot^2(\theta) = \csc^2(\theta)\]

5. Compound Angle Formulas (Sum & Difference)

Compound angle relations allow the expansion and simplification of trigonometric functions containing angle sums or differences:

\[\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\] \[\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\] \[\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}\]

6. Double Angle & Power-Reduction Formulas

Double angle identities and power-reduction identities simplify expressions involving higher powers or multiples of angles:

\[\sin(2A) = 2 \sin A \cos A\] \[\cos(2A) = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A\] \[\tan(2A) = \frac{2\tan A}{1 - \tan^2 A}\] \[\sin^2(A) = \frac{1 - \cos(2A)}{2}, \quad \cos^2(A) = \frac{1 + \cos(2A)}{2}\]

7. Product-to-Sum Identities

Product-to-sum transformations convert product terms of sine and cosine into additions or subtractions, making integration and simplification straightforward:

\[\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]\] \[\cos A \sin B = \frac{1}{2}[\sin(A+B) - \sin(A-B)]\] \[\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]\] \[\sin A \sin B = -\frac{1}{2}[\cos(A+B) - \cos(A-B)]\]

8. Laws of Sines and Cosines for General Triangles

For any oblique triangle with side lengths \(a, b, c\) opposite to angles \(A, B, C\) respectively, the following rules apply:

Law of Sines

\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

Law of Cosines

\[c^2 = a^2 + b^2 - 2ab \cos C\]

9. Frequently Asked Questions (FAQ)

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