Trigonometry has more formulas than any other Class 11 chapter, but they come in families, and each family answers one kind of question. This sheet is arranged by family, with when to reach for it beside each block.
Ratios and the standard values
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = sin θ/cos θ; cosec, sec, cot are the reciprocals.
| θ | 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 | undefined |
Radians: 180° = π, so 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2. Arc length l = rθ (θ in radians).
The fundamental identities — for simplifying
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
Use when an expression mixes squares of ratios. Worked line: 1 − cos²θ = sin²θ, so (1 − cos²θ)/sin θ = sin θ.
Signs and related angles — for any angle
Quadrant rule (All, Sine, Tan, Cos positive in I, II, III, IV).
- sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ
- sin(90° − θ) = cos θ, cos(90° − θ) = sin θ
- sin(180° − θ) = sin θ, cos(180° − θ) = −cos θ
- sin(180° + θ) = −sin θ, cos(180° + θ) = −cos θ
- sin(360° − θ) = −sin θ, cos(360° − θ) = cos θ
Rule of thumb: with 90° or 270° the ratio changes (sin ↔ cos); with 180° or 360° it stays; the sign comes from the quadrant. Worked line: cos 210° = cos(180° + 30°) = −cos 30° = −√3/2.
Compound angles — for sums and differences
- 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)
Use when the angle is a sum of standard angles (15°, 75°, 105°). Worked line: cos 75° = cos(45° + 30°) = (1/√2)(√3/2) − (1/√2)(1/2) = (√3 − 1)/(2√2).
Multiple angles — for 2θ and 3θ
- sin 2θ = 2 sin θ cos θ = 2 tan θ/(1 + tan²θ)
- cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ = (1 − tan²θ)/(1 + tan²θ)
- tan 2θ = 2 tan θ/(1 − tan²θ)
- sin 3θ = 3 sin θ − 4 sin³θ
- cos 3θ = 4 cos³θ − 3 cos θ
- tan 3θ = (3 tan θ − tan³θ)/(1 − 3 tan²θ)
The three forms of cos 2θ are the most-used lines in the chapter. Use when you need to halve or double an angle. Worked line: cos²θ = (1 + cos 2θ)/2 turns a square into a single ratio — essential later in integration.
Sum to product — for sums of two ratios
- sin A + sin B = 2 sin[(A + B)/2] cos[(A − B)/2]
- sin A − sin B = 2 cos[(A + B)/2] sin[(A − B)/2]
- cos A + cos B = 2 cos[(A + B)/2] cos[(A − B)/2]
- cos A − cos B = −2 sin[(A + B)/2] sin[(A − B)/2]
Use when proving identities like (sin 5x + sin 3x)/(cos 5x + cos 3x) = tan 4x. Worked line: numerator = 2 sin 4x cos x; denominator = 2 cos 4x cos x; ratio = tan 4x.
Product to sum — the reverse
- 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)
Use when a product needs to become something you can integrate or evaluate.
General solutions — for solving equations
- sin θ = 0 ⇒ θ = nπ; cos θ = 0 ⇒ θ = (2n + 1)π/2; tan θ = 0 ⇒ θ = nπ
- sin θ = sin α ⇒ θ = nπ + (−1)ⁿα
- cos θ = cos α ⇒ θ = 2nπ ± α
- tan θ = tan α ⇒ θ = nπ + α
(n any integer.) Worked line: 2 sin θ = 1 ⇒ sin θ = sin(π/6) ⇒ θ = nπ + (−1)ⁿ π/6.
The three you will mix up
- cos(A + B) has a minus in the middle; sin(A + B) has a plus. Say "cos changes sign".
- cos A − cos B has a leading minus in sum-to-product; the other three do not.
- The general solution of sin carries (−1)ⁿ; cos carries ±; tan carries neither.