Trigonometric Ratios & Identities
For IIT‑JEE (Main & Advanced) – Mathematics
1. Simple Explanation
Trigonometry studies the relationships between the angles and sides of triangles (mostly right‑angled) and the periodic functions that arise from the unit circle.
In JEE you’ll mainly use:
| Ratio | Definition (for a right‑angled triangle) | Unit‑circle expression |
|---|---|---|
| sin θ | opposite / hypotenuse | y / r |
| cos θ | adjacent / hypotenuse | x / r |
| tan θ | opposite / adjacent | y / x |
| cot θ | adjacent / opposite | x / y |
| sec θ | hypotenuse / adjacent | r / x |
| csc θ | hypotenuse / opposite | r / y |
θis the angle measured from the positive x-axis.- On the unit circle,
r = 1, so the coordinates of the point are(cos θ, sin θ).
2. Key Points & Important Identities
| Category | Formula | Notes |
|---|---|---|
| Pythagorean Identities | sin²θ + cos²θ = 1 |
Base identity. |
1 + tan²θ = sec²θ |
Derived by dividing the first by cos²θ. |
|
1 + cot²θ = csc²θ |
Derived by dividing the first by sin²θ. |
|
| Reciprocal Identities | sinθ = 1 / cscθ cosθ = 1 / secθ tanθ = 1 / cotθ |
Useful for simplifying. |
| Co‑function Identities | sin(π/2 – θ) = cosθ cos(π/2 – θ) = sinθ tan(π/2 – θ) = cotθ |
Convert between functions. |
| Sum & Difference | sin(A ± B) = sinA cosB ± cosA sinB |
|
cos(A ± B) = cosA cosB ∓ sinA sinB |
||
tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB) |
||
| Double & Half Angles | sin 2θ = 2 sinθ cosθ cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ tan 2θ = 2 tanθ / (1 – tan²θ) |
|
sin(θ/2) = ±√[(1 – cosθ)/2] cos(θ/2) = ±√[(1 + cosθ)/2] |
Sign chosen by quadrant. | |
| Product‑to‑Sum | sinA sinB = ½[cos(A–B) – cos(A+B)] cosA cosB = ½[cos(A–B) + cos(A+B)] sinA cosB = ½[sin(A+B) + sin(A–B)] |
Helps in integration & simplifying expressions. |
| Sum‑to‑Product | sinA + sinB = 2 sin[(A+B)/2] cos[(A–B)/2] sinA – sinB = 2 cos[(A+B)/2] sin[(A–B)/2] cosA + cosB = 2 cos[(A+B)/2] cos[(A–B)/2] cosA – cosB = –2 sin[(A+B)/2] sin[(A–B)/2] |
Reverse of product‑to‑sum. |
| Power‑Reduction | sin²θ = ½[1 – cos2θ] cos²θ = ½[1 + cos2θ] |
Useful for integrating powers of sin/cos. |
| Sum/Difference of Angles in Quadratic | sin²θ + sin²(α–θ) = sin²α |
Appears in geometry & optimization problems. |
| Angle Increments | sin(θ + 2π) = sinθ, cos(θ + 2π) = cosθ |
Periodicity. |
| Special Angles | sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3 etc. |
Memorise 0°, 30°, 45°, 60°, 90°, 180°, 270°, 360°. |
Common Mistakes
- Forgetting the sign of
sin(θ/2)orcos(θ/2)→ always check the quadrant.- Mixing up the “±” signs in sum/difference identities.
- Assuming
tan(A+B) = tanA + tanB– the denominator term is crucial.
3. Worked Example
Problem (IIT‑JEE style)
Evaluate
[ \frac{\sin 20^\circ \cos 40^\circ + \cos 20^\circ \sin 40^\circ}{\sin 20^\circ \cos 20^\circ} ]
Step‑by‑Step Solution
- Recognise the numerator
[ \sin 20^\circ \cos 40^\circ + \cos 20^\circ \sin 40^\circ = \sin(20^\circ + 40^\circ) \quad \text{(sum identity)} = \sin