Trigonometry Identities Class 11: Complete Formula List with Memory Tricks

Trigonometric Functions (Class 11 Maths, Chapter 3) has more formulas than any other chapter, and they return in Class 12 integration and in JEE. This post lists every identity with a trick to remember it.

1. Basic identities

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

2. Standard values

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	∞

Trick: sin values are √0/2, √1/2, √2/2, √3/2, √4/2. Reverse for cos.

3. Allied angles (sign rule)

“All Students Take Coffee”: Quadrant I all positive, II only sin (and cosec), III only tan (and cot), IV only cos (and sec). For 90° ± θ and 270° ± θ the function changes (sin ↔ cos, tan ↔ cot); for 180° ± θ and 360° ± θ it stays the same.

4. Compound angles

• sin(A ± B) = sinA cosB ± cosA sinB
• cos(A ± B) = cosA cosB ∓ sinA sinB
• tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB)

5. Multiple and half angles

• sin2A = 2sinA cosA = 2tanA/(1 + tan²A)
• cos2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A = (1 − tan²A)/(1 + tan²A)
• tan2A = 2tanA/(1 − tan²A)
• sin3A = 3sinA − 4sin³A; cos3A = 4cos³A − 3cosA
• 1 + cos2A = 2cos²A; 1 − cos2A = 2sin²A (used in integration)

6. Sum to product

• sinC + sinD = 2 sin((C+D)/2) cos((C−D)/2)
• sinC − sinD = 2 cos((C+D)/2) sin((C−D)/2)
• cosC + cosD = 2 cos((C+D)/2) cos((C−D)/2)
• cosC − cosD = −2 sin((C+D)/2) sin((C−D)/2)

7. Product to sum

• 2sinA cosB = sin(A+B) + sin(A−B)
• 2cosA sinB = sin(A+B) − sin(A−B)
• 2cosA cosB = cos(A+B) + cos(A−B)
• 2sinA sinB = cos(A−B) − cos(A+B)

Solved examples

1. Find sin 75°. sin(45° + 30°) = (1/√2)(√3/2) + (1/√2)(1/2) = (√3 + 1)/2√2.

2. Prove (sin3A)/(sinA) − (cos3A)/(cosA) = 2. LHS = (sin3A cosA − cos3A sinA)/(sinA cosA) = sin(3A − A)/(sinA cosA) = sin2A/(sinA cosA) = 2. ✓

Common mistakes

• Sign of cos(A + B) — it is minus.
• Forgetting the 1/2 in sum-to-product angles.
• Mixing radians and degrees (π = 180°).

These identities feed directly into Class 12 integration.

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Frequently asked questions

How do I memorise all trigonometry formulas?

Learn them in groups (basic, compound, multiple, sum-product), write each group daily for a week, and derive product-to-sum from compound angles instead of memorising.

What is the value of sin 15°?

(√3 − 1)/2√2, from sin(45° − 30°).

Which formulas are used most in Class 12?

1 ± cos2A = 2cos²A / 2sin²A, product-to-sum formulas and sin2A, cos2A, mainly in integration.