⭐ Sabse Important Points · एक नज़र में
- Pythagorean Identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ. These are foundational for simplifying expressions.
- Sine Rule: a/sinA = b/sinB = c/sinC = 2R (R = circumradius). Used when two angles and one side or two sides and opposite angle are known.
- Allied Angles Rule: For angles of form nπ/2 ± θ, if n is even, function remains same; if n is odd, function changes to co-function. Sign depends on quadrant of original angle.
- Cosine Rule: a² = b² + c² - 2bc cosA. Used when three sides or two sides and included angle are known.
- Sum and Difference Formulas: sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB; tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB).
- Inverse Functions: sin⁻¹x has range [-π/2, π/2]; cos⁻¹x has range [0, π]; tan⁻¹x has range (-π/2, π/2). Principal value is unique.
Introduction · परिचय
Trigonometry is a core topic in NDA Mathematics, covering ratios, identities, equations, and properties of triangles. It tests both conceptual understanding and computational speed. Key areas include trigonometric ratios of standard and allied angles, identities, inverse functions, and solution of triangles. Mastery of formulas and their applications is essential for scoring well.
Trigonometric Ratios and Identities
Trigonometric ratios relate angles to side lengths in a right triangle. Fundamental identities like Pythagorean, quotient, and reciprocal identities are used to simplify expressions and solve equations. Allied angles (e.g., 90°±θ, 180°±θ) follow specific sign and co-function rules.
📐 Formula / Rule Box · सूत्र याद रखो
Pythagorean Identity
sin²θ + cos²θ = 1
Sum Formula for Sine
sin(A+B) = sinA cosB + cosA sinB
Double Angle for Cosine
cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
Allied Angles Rule
For angles of form nπ/2 ± θ, if n is even, function remains same; if n is odd, function changes to co-function. Sign depends on quadrant of original angle.
Sum and Difference Formulas
sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB; tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB).
⭐ Key Points · याद रखने वाली बातें
- Pythagorean Identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ. These are foundational for simplifying expressions.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find the value of sin 75°.
✅ Solution: sin 75° = (√6 + √2)/4
💡 Explanation: Use sin(45°+30°) = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.
Solution of Triangles and Inverse Trigonometric Functions
Solution of triangles involves sine rule, cosine rule, and projection formula to find unknown sides/angles. Inverse trigonometric functions have restricted domains and ranges; principal values are essential.
📐 Formula / Rule Box · सूत्र याद रखो
Sine Rule
a/sinA = b/sinB = c/sinC = 2R
Cosine Rule
a² = b² + c² - 2bc cosA
Projection Formula
a = b cosC + c cosB
Sine Rule
a/sinA = b/sinB = c/sinC = 2R (R = circumradius). Used when two angles and one side or two sides and opposite angle are known.
Cosine Rule
a² = b² + c² - 2bc cosA. Used when three sides or two sides and included angle are known.
| Sine Rule | Cosine Rule |
|---|---|
| Used when two angles and one side known | Used when three sides or two sides and included angle known |
| Involves ratio of side to sine of opposite angle | Relates all three sides to cosine of one angle |
| Simpler for finding unknown side given two angles | Necessary when only sides are given |
⭐ Key Points · याद रखने वाली बातें
- Inverse Functions: sin⁻¹x has range [-π/2, π/2]; cos⁻¹x has range [0, π]; tan⁻¹x has range (-π/2, π/2). Principal value is unique.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: In triangle ABC, a=7, b=8, c=9. Find angle C.
✅ Solution: C = cos⁻¹(2/7) ≈ 73.4°
💡 Explanation: Using cosine rule: cosC = (a²+b²-c²)/(2ab) = (49+64-81)/(2*7*8) = 32/112 = 2/7. Thus C = cos⁻¹(2/7).
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1If sinθ + cosθ = √2, find the value of sinθ cosθ.
👀 Answer dekho
✅ Answer: 1/2
💡 Explanation: Square both sides: (sinθ+cosθ)² = 2 ⇒ sin²θ + cos²θ + 2sinθcosθ = 2 ⇒ 1 + 2sinθcosθ = 2 ⇒ 2sinθcosθ = 1 ⇒ sinθcosθ = 1/2.
Q2Find the principal value of tan⁻¹(√3).
👀 Answer dekho
✅ Answer: π/3
💡 Explanation: tan(π/3) = √3 and π/3 lies in (-π/2, π/2), the range of tan⁻¹. Hence principal value is π/3.
Q3In triangle ABC, if a=5, b=7, and C=60°, find side c.
👀 Answer dekho
✅ Answer: √39
💡 Explanation: Using cosine rule: c² = a² + b² - 2ab cosC = 25 + 49 - 2*5*7*(1/2) = 74 - 35 = 39 ⇒ c = √39.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Forgetting to check the quadrant when using allied angles, leading to wrong sign. Always determine the quadrant of the original angle.
- Applying sine rule when two sides and included angle are given (should use cosine rule). Remember sine rule needs a side and its opposite angle.
- Ignoring the restricted range of inverse trigonometric functions, giving a value outside the principal range.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Standard Values: Memorize sin, cos, tan for 0°, 30°, 45°, 60°, 90° to speed up calculations.
- Identities: sin²θ + cos²θ = 1; sec²θ - tan²θ = 1; cosec²θ - cot²θ = 1 are used to simplify expressions.
- Sine Rule: a/sinA = b/sinB = c/sinC = 2R; use when two angles and one side or two sides and opposite angle are known.
- Cosine Rule: a² = b² + c² - 2bc cosA; use when three sides or two sides and included angle are known.
- Inverse Ranges: sin⁻¹x ∈ [-π/2, π/2], cos⁻¹x ∈ [0, π], tan⁻¹x ∈ (-π/2, π/2); principal values are unique.