📐 Trigonometry & Identities: Formulas, Values & Proof Strategies

📑 In This Chapter Guide (Table of Contents)

1. 1. Trigonometric Ratios (T-Ratios) in a Right Triangle
2. 2. Standard Trigonometric Values Table (0° to 90°)
3. 3. Fundamental Identities & Top Proving Strategies
4. 💡 Frequently Asked Questions (FAQ)

1. Trigonometric Ratios (T-Ratios) in a Right Triangle

For a right-angled triangle ΔABC with right angle at B and acute angle θ = ∠A:

• sin θ = Opposite Side (Perpendicular) / Hypotenuse = P / H
• cos θ = Adjacent Side (Base) / Hypotenuse = B / H
• tan θ = Opposite / Adjacent = P / B = sin θ / cos θ
• cosec θ = 1 / sin θ = H / P
• sec θ = 1 / cos θ = H / B
• cot θ = 1 / tan θ = B / P = cos θ / sin θ

2. Standard Trigonometric Values Table (0° to 90°)

3. Fundamental Identities & Top Proving Strategies

The 3 Fundamental Identities:

1. sin²θ + cos²θ = 1 ⟹ sin²θ = 1 - cos²θ ⟹ cos²θ = 1 - sin²θ
2. 1 + tan²θ = sec²θ ⟹ sec²θ - tan²θ = 1 ⟹ (sec θ - tan θ)(sec θ + tan θ) = 1
3. 1 + cot²θ = cosec²θ ⟹ cosec²θ - cot²θ = 1 ⟹ (cosec θ - cot θ)(cosec θ + cot θ) = 1

4 Rules of Thumb for Identity Proof Questions:

• Convert everything into terms of sin θ and cos θ when you are stuck.
• Take the Lowest Common Denominator (LCM) of fractional terms.
• Rationalize the numerator or denominator when terms like (1 ± sin θ) or (1 ± cos θ) appear inside square roots.
• Apply algebraic identities: a² - b² = (a - b)(a + b), a³ + b³ = (a + b)(a² - ab + b²).

💡 Frequently Asked Questions (FAQ)

📚 Related Study Guides & Notes

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