Standard Trigonometric Values
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sinθ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cosθ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tanθ | 0 | 1/√3 | 1 | √3 | undef |
| cosecθ | undef | 2 | √2 | 2/√3 | 1 |
| secθ | 1 | 2/√3 | √2 | 2 | undef |
| cotθ | undef | √3 | 1 | 1/√3 | 0 |
This cheat sheet contains all formulas essential for solving Type 1 (Short Questions), Type 2 (Long Questions), and Type 3 (Conditional Proofs) in your SEE Optional Math board examinations.
1. Multiple Angles (2A and 3A)
Formulas for Sine (sin 2A, sin 3A)
- 🔹 sin 2A = 2 sin A · cos A
- 🔹 sin 2A = 2 tan A1 + tan² A
- 🔹 sin 3A = 3 sin A - 4 sin³ A → (Hence, 4 sin³ A = 3 sin A - sin 3A)
Formulas for Cosine (cos 2A, cos 3A)
- 🔹 cos 2A = cos² A - sin² A
- 🔹 cos 2A = 2 cos² A - 1 → 1 + cos 2A = 2 cos² A (Highly Used!)
- 🔹 cos 2A = 1 - 2 sin² A → 1 - cos 2A = 2 sin² A (Highly Used!)
- 🔹 cos 2A = 1 - tan² A1 + tan² A
- 🔹 cos 3A = 4 cos³ A - 3 cos A → (Hence, 4 cos³ A = 3 cos A + cos 3A)
Formulas for Tangent (tan 2A, tan 3A)
- 🔹 tan 2A = 2 tan A1 - tan² A
- 🔹 tan 3A = 3 tan A - tan³ A1 - 3 tan² A
2. Sub-Multiple Angles (A/2 and A/3)
Note: These are identical to multiple angles, but replace angle "A" with "A/2" or "A/3".
Formulas for sin A, cos A, and tan A
- 🔹 sin A = 2 sin(A/2) · cos(A/2) or 2 tan(A/2)1 + tan²(A/2)
- 🔹 cos A = cos²(A/2) - sin²(A/2)
- 🔹 cos A = 2 cos²(A/2) - 1 → 1 + cos A = 2 cos²(A/2)
- 🔹 cos A = 1 - 2 sin²(A/2) → 1 - cos A = 2 sin²(A/2)
- 🔹 cos A = 1 - tan²(A/2)1 + tan²(A/2)
- 🔹 tan A = 2 tan(A/2)1 - tan²(A/2)
3. Transformation of Products & Sums
Product into Sum or Difference Form (2 Product Rules)
- 🔹 2 sin A · cos B = sin(A + B) + sin(A - B)
- 🔹 2 cos A · sin 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)
⚠️ Be Careful: In the 4th formula (2 sin A · sin B), the negative term (A - B) comes first!
Sum or Difference into Product Form (C & D Formulae)
- 🔹 sin C + sin D = 2 sinC + D2· cosC - D2
- 🔹 sin C - sin D = 2 cosC + D2· sinC - D2
- 🔹 cos C + cos D = 2 cosC + D2· cosC - D2
- 🔹 cos C - cos D = 2 sinC + D2· sinD - C2
⚠️ Be Careful: In the final formula (cos C - cos D), the last part contains (D - C)/2 instead of (C - D)/2!
4. Conditional Trigonometric Identities
When given the constraint: A + B + C = 180° (πc)
Angle Conversions Rules
- 🔹 sin(A + B) = sin C 🔹 sin(A + B2) = cos(C2)
- 🔹 cos(A + B) = -cos C 🔹 cos(A + B2) = sin(C2)
- 🔹 tan(A + B) = -tan C 🔹 tan(A + B2) = cot(C2)