NDA formula sheet for Trigonometry! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Chapter Summary: Trigonometry Key Formulas & Concepts

Trigonometry is a foundational topic for the National Defence Academy (NDA) mathematics examination. Mastery over fundamental identities, compound angles, and multiple-angle transformation formulas is critical for solving questions efficiently during the test.

1. Basic Trigonometric Identities

For any angle \(\theta\), the fundamental pythagorean identities form the backbone of trigonometric simplifications:

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

2. Compound & Double Angle Formulas

Addition and subtraction of angles allow decomposition of complex expression components:

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

Expressing double angles in terms of single angles is equally imperative for NDA problem-solving:

\[\sin 2\theta = 2\sin\theta\cos\theta = \frac{2\tan\theta}{1+\tan^2\theta}\]\[\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta\]

3. Transformation Formulas

Converting product forms into sums or differences simplifies definite integrals and algebraic reductions:

• \(2\sin A \cos B = \sin(A+B) + \sin(A-B)\)
• \(2\cos A \cos B = \cos(A+B) + \cos(A-B)\)
• \(2\sin A \sin B = \cos(A-B) - \cos(A+B)\)

4. Properties of Triangles

In any triangle \(ABC\) with sides \(a, b, c\) opposite to angles \(A, B, C\):

Sine Rule: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\) (where \(R\) is the circumradius).

Cosine Rule: \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)