Trigonometry extends beyond triangles to the study of periodic phenomena. In Class 11, we generalise trigonometric ratios (defined for acute angles in right triangles) to trigonometric functions defined for all real numbers.
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Angle Measurement
- An angle is formed by rotating a ray about its initial position. Angles can be measured in:
- Degrees: One full rotation = 360°.
- Radians: One full rotation = 2π radians.
Conversion: Degrees × π/180 = Radians; Radians × 180/π = Degrees.
Arc length formula: l = r × θ (where θ is in radians).
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Trigonometric Functions: Definition
- For a point P(x, y) on a circle of radius r centred at origin, the six trig functions are:
- sin θ = y/r, cos θ = x/r, tan θ = y/x (x ≠ 0)
- cosec θ = r/y (y ≠ 0), sec θ = r/x (x ≠ 0), cot θ = x/y (y ≠ 0)
Signs in quadrants (ASTC rule): All positive (I), Sin positive (II), Tan positive (III), Cos positive (IV).
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Key Values Table
| Angle | 0° | 30° | 45° | 60° | 90° |
|------------|-----|-------|--------|-------|-----|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | — |
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Important Identities
- Pythagorean identities:
- sin2 θ + cos2 θ = 1
- 1 + tan2 θ = sec2 θ
- 1 + cot2 θ = cosec2 θ
- Sum and difference formulas:
- sin(A + B) = sin A cos B + cos A sin B
- cos(A + B) = cos A cos B − sin A sin B
- tan(A + B) = (tan A + tan B)/(1 − tan A tan B)
- Double angle formulas:
- sin 2A = 2 sin A cos A
- cos 2A = cos2 A − sin2 A = 2cos2 A − 1 = 1 − 2sin2 A
- tan 2A = 2 tan A/(1 − tan2 A)
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Worked Examples
Convert 210° to radians.
210 × π/180 = 7π/6 radians.
Prove: (1 − sin2 θ)/cos θ = cos θ.
LHS = cos2 θ / cos θ = cos θ = RHS.
Find sin 75°.
sin 75° = sin(45° + 30°) = sin45·cos30 + cos45·sin30 = (1/√2)(√3/2) + (1/√2)(1/2) = (√3 + 1)/(2√2).
If sin A = 3/5 and A is in the first quadrant, find cos A and tan A.
cos2 A = 1 − 9/25 = 16/25, so cos A = 4/5. tan A = (3/5)/(4/5) = 3/4.
Prove: sin 2A = 2 sin A cos A using the sum formula.
sin 2A = sin(A + A) = sin A cos A + cos A sin A = 2 sin A cos A.
Find the value of cos 120°.
cos 120° = cos(180° − 60°) = −cos 60° = −1/2.
Simplify: cos2 θ/(1 − tan2 θ).
cos2 θ/(1 − sin2 θ/cos2 θ) = cos4 θ/(cos2 θ − sin2 θ) = cos4 θ/cos 2θ.
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Common mistakes
- Forgetting the ASTC sign rule when applying functions in different quadrants.
- Using sin(A + B) = sin A + sin B (WRONG). Always use the correct compound angle formula.
- Confusing radian and degree measure — always check which unit is being used.
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Summary
Trigonometric functions extend trig ratios to all real values. The key identities — Pythagorean, compound angle, double angle — allow us to simplify and evaluate expressions efficiently.