1. Primary Trigonometric Ratios
In a right-angled triangle with an acute angle \(\theta\), the fundamental trigonometric ratios relate the lengths of the sides to the interior angle. Let side \(a\) be the Opposite side, side \(b\) be the Adjacent side, and side \(c\) be the Hypotenuse.
The primary ratios and their reciprocals are defined as follows:
\[\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{a}{c} \quad \vert \quad \csc\theta = \frac{1}{\sin\theta} = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{c}{a}\] \[\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{b}{c} \quad \vert \quad \sec\theta = \frac{1}{\cos\theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{c}{b}\] \[\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin\theta}{\cos\theta} = \frac{a}{b} \quad \vert \quad \cot\theta = \frac{1}{\tan\theta} = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{b}{a}\]2. Fundamental Pythagorean Identities
The fundamental Pythagorean identities form the bedrock of trigonometric algebra for the UPSC CDS exam. These identities hold true for all real angles \(\theta\) where the trigonometric functions are defined.
\[\sin^2\theta + \cos^2\theta = 1\] \[1 + \tan^2\theta = \sec^2\theta\] \[1 + \cot^2\theta = \csc^2\theta\]Note for CDS Aspirants: Algebraic variations such as \(\sec^2\theta - \tan^2\theta = 1\) and \(\csc^2\theta - \cot^2\theta = 1\) are frequently used in simplifying complex questions involving algebraic factorization like \((\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1\).
3. Angle Sum and Difference Formulas
Compound angle identities allow you to evaluate trigonometric functions at sums or differences of angles \(\alpha\) and \(\beta\):
\[\sin(\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha \sin\beta\] \[\cos(\alpha \pm \beta) = \cos\alpha \cos\beta \mp \sin\alpha \sin\beta\] \[\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha \tan\beta}\]4. Double-Angle and Half-Angle Formulas
Double-Angle Formulas
Expressing trigonometric functions of \(2\theta\) in terms of single-angle functions \(\theta\):
\[\sin(2\theta) = 2\sin\theta\cos\theta\] \[\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta\] \[\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}\]Half-Angle Formulas
Derivatives used to compute half-angle values \(\frac{\theta}{2}\):
\[\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos\theta}{2}}\] \[\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos\theta}{2}}\] \[\tan(\frac{\theta}{2}) = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}\]5. CDS 2027 Exam Analysis & Chapter Weightage
PYQ Analysis & NTA/UPSC Trend
In the UPSC Combined Defence Services (CDS) Mathematics paper, Trigonometry carries a consistent weightage of 12% to 15% (approximately 12 to 15 questions out of 100). The questions test both standard application of identities and conceptual transformation of complex ratios.
Chapter Weightage Breakdown
- Basic Trigonometric Ratios & Identities: 4–5 Questions
- Heights and Distances: 3–4 Questions
- Trigonometric Equations & Compound Angles: 3–4 Questions
- Maximum and Minimum Values of Trigonometric Functions: 1–2 Questions
6. Frequently Asked Questions (People Also Ask)
Q1: How many questions are asked from Trigonometry in the CDS exam?
Typically, around 12 to 15 questions appear in the UPSC CDS Elementary Mathematics paper, making Trigonometry one of the highest-yield chapters along with Mensuration and Geometry.
Q2: What is the primary identity used to solve CDS height and distance questions?
Heights and Distances questions heavily rely on the primary ratios \(\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}\) and \(\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\), especially for standard angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\).
Q3: Are half-angle formulas important for CDS 2027?
Yes, half-angle formulas are directly useful for simplifying radical expressions and finding values for non-standard angles such as \(15^\circ\) or \(22.5^\circ\) in direct CDS PYQs.
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