CDS 2027 lesson summary for Trigonometry! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Chapter Summary: Trigonometry

Trigonometry is a crucial module for the Combined Defence Services (CDS) examination, testing a candidate's grasp of angles, geometric identities, and spatial measurements.

1. Fundamental Trigonometric Ratios & Identities

For a right-angled triangle with angle \(\theta\), base \(B\), perpendicular \(P\), and hypotenuse \(H\):

• Primary Ratios: \(\sin\theta = \frac{P}{H}\), \(\cos\theta = \frac{B}{H}\), \(\tan\theta = \frac{P}{B}\)
• Reciprocal Ratios: \(\csc\theta = \frac{1}{\sin\theta}\), \(\sec\theta = \frac{1}{\cos\theta}\), \(\cot\theta = \frac{1}{\tan\theta}\)

Pythagorean Identities

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

2. Angle Ratios & Quadrant Rules

Signs of trigonometric functions depend on the quadrant location of angle \(\theta\):

• Quadrant I (0 to \(\frac{\pi}{2}\)): All ratios are positive.
• Quadrant II (\(\frac{\pi}{2}\) to \(\pi\)): Only \(\sin\theta\) and \(\csc\theta\) are positive.
• Quadrant III (\(\pi\) to \(\frac{3\pi}{2}\)): Only \(\tan\theta\) and \(\cot\theta\) are positive.
• Quadrant IV (\(\frac{3\pi}{2}\) to \(2\pi\)): Only \(\cos\theta\) and \(\sec\sec\theta\) are positive.

3. Compound & Multiple Angle Formulas

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

4. Heights and Distances

Applications in geometric navigation rely on determining angles of elevation and depression using basic ratios: \(\tan\theta = \frac{\text{Height}}{\text{Distance}}\).