Trigonometry Cheat Sheet & Formula Guide - NDA Exam

Welcome to this comprehensive LibreTexts-style study module designed specifically for NDA (National Defence Academy) aspirants. Trigonometry is one of the highest-weightage topics in the NDA Mathematics paper, typically accounting for 12 to 15 questions (30 to 37.5 marks). This chapter breaks down fundamental right-angled triangle definitions, core Pythagorean identities, standard angle tables, and compound angle formulas with expert analysis tailored to the latest examination pattern.

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1. Right-Angled Triangle Definitions & Ratios

Trigonometric ratios form the foundation of geometry and calculus in the NDA syllabus. In a right-angled triangle with sides labeled as Opposite (\(a\)), Adjacent (\(b\)), and Hypotenuse (\(c\)) relative to an angle \(\theta\):

• Sine (\(\sin\theta\)): Ratio of Opposite to Hypotenuse \[\sin\theta = \frac{\text{Opp}}{\text{Hyp}} = \frac{a}{c}\]
• Cosine (\(\cos\theta\)): Ratio of Adjacent to Hypotenuse \[\cos\theta = \frac{\text{Adj}}{\text{Hyp}} = \frac{b}{c}\]
• Tangent (\(\tan\theta\)): Ratio of Opposite to Adjacent \[\tan\theta = \frac{\text{Opp}}{\text{Adj}} = \frac{a}{b}\]
• Cosecant (\(\csc\theta\)): Ratio of Hypotenuse to Opposite \[\csc\theta = \frac{\text{Hyp}}{\text{Opp}} = \frac{c}{a}\]
• Secant (\(\sec\theta\)): Ratio of Hypotenuse to Adjacent \[\sec\theta = \frac{\text{Hyp}}{\text{Adj}} = \frac{c}{b}\]
• Cotangent (\(\cot\theta\)): Ratio of Adjacent to Opposite \[\cot\theta = \frac{\text{Adj}}{\text{Opp}} = \frac{b}{a}\]

2. Fundamental Identities

Understanding the interplay between trigonometric ratios allows rapid simplification of complex expressions under timed NDA exam conditions.

Reciprocal Identities

\[\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}\]

Quotient Identities

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

Pythagorean Identities

Derived directly from the Pythagorean theorem (\(a^2 + b^2 = c^2\)):

\[\sin^2\theta + \cos^2\theta = 1\] \[1 + \tan^2\theta = \sec^2\theta \implies \sec^2\theta - \tan^2\theta = 1\] \[1 + \cot^2\theta = \csc^2\theta \implies \csc^2\theta - \cot^2\theta = 1\]

3. Table of Standard Trigonometric Values

Memorizing these exact values in both degrees and radians is crucial for solving height and distance problems as well as evaluating definite integrals in calculus.

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

4. Compound Angle Formulas

Compound angle identities expand trigonometric functions applied to sums or differences of angles \(A\) and \(B\):

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

5. NDA Exam Analysis & Preparation Strategy

PYQ Analysis & Chapter Weightage

Historical trends in UPSC NDA papers show that Trigonometry questions test both memory of identities and algebraic manipulation skills. Expect questions combining compound angles with quadratic equations or inverse trigonometric functions.

Exam Pattern Insights

• Speed Trick: Substitute options like \(\theta = 30^\circ\) or \(\theta = 45^\circ\) to eliminate incorrect choices rapidly without full algebraic expansion.
• Common Traps: Be careful with sign changes in \(\cos(A + B)\) vs \(\cos(A - B)\), where the middle operator flips.

Frequently Asked Questions (People Also Ask)