CDS 2027 - Paper 3: Elementary Mathematics cheat sheet for Trigonometry! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Chapter Summary: Elementary Trigonometry

Trigonometry in the CDS Elementary Mathematics syllabus focuses on right-angled triangle properties, trigonometric identities, circular measure of angles, and practical applications in heights and distances.

1. Trigonometric Ratios & Identities

For a right-angled triangle with angle \(\theta\), the primary ratios are defined as:

• \(\sin\theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}}\), \(\cos\theta = \frac{\text{Base}}{\text{Hypotenuse}}\), \(\tan\theta = \frac{\text{Perpendicular}}{\text{Base}}\)
• \(\csc\theta = \frac{1}{\sin\theta}\), \(\sec\theta = \frac{1}{\cos\theta}\), \(\cot\theta = \frac{1}{\tan\theta}\)

Fundamental Pythagorean Identities:

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

2. Complementary Angles & Values

Ratios of complementary angles satisfy key relationships:

• \(\sin(90^\circ - \theta) = \cos\theta\)
• \(\tan(90^\circ - \theta) = \cot\theta\)
• \(\sec(90^\circ - \theta) = \csc\theta\)

Standard values for angles \(0^\circ, 30^\circ, 45^\circ, 60^\circ,\) and \(90^\circ\) must be memorized for speed. For example, \(\sin 30^\circ = \frac{1}{2}\), \(\cos 45^\circ = \frac{1}{\sqrt{2}}\), and \(\tan 60^\circ = \sqrt{3}\).

3. Heights and Distances

Problems involve visual angles measured from an observer's line of sight:

• Angle of Elevation: Measured upward from the horizontal ground to an object above.
• Angle of Depression: Measured downward from a horizontal line to an object below.

Key relation for a tower of height \(h\) observed at distance \(d\) with elevation angle \(\theta\): \[\tan\theta = \frac{h}{d}\]