Trigonometry : Formulas, Concepts & Exam-Saving Tricks
Trigonometry is about understanding the relationship between angles and sides of a triangle. Once the core patterns become familiar, most problems can be solved with just a handful of identities and smart shortcuts.
1. The Foundation: What Are Sine, Cosine and Tangent?
For a right-angled triangle:
| Function | Definition | Easy Memory |
|---|---|---|
| sin θ | Perpendicular / Hypotenuse | Opposite / Hypotenuse |
| cos θ | Base / Hypotenuse | Adjacent / Hypotenuse |
| tan θ | Perpendicular / Base | Opposite / Adjacent |
Quick Memory Trick: SOH-CAH-TOA
- SOH → Sin = Opposite / Hypotenuse
- CAH → Cos = Adjacent / Hypotenuse
- TOA → Tan = Opposite / Adjacent
2. The Most Important Standard Values
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | Undefined |
Exam Tip: Most trigonometry questions eventually reduce to these five angles.
3. Reciprocal Functions
| Function | Reciprocal |
|---|---|
| sin θ | cosec θ = 1 / sin θ |
| cos θ | sec θ = 1 / cos θ |
| tan θ | cot θ = 1 / tan θ |
Whenever you see sec, cosec or cot, convert them into sine and cosine first. Most simplifications become much easier.
4. The Three Golden Identities
These identities appear everywhere in trigonometry.
Identity 1
sin²θ + cos²θ = 1
Identity 2
1 + tan²θ = sec²θ
Identity 3
1 + cot²θ = cosec²θ
Shortcut: If you remember the first identity, the other two can be derived whenever needed.
5. Complementary Angle Relationships
When two angles add up to 90°, they are complementary.
| Expression | Equivalent |
|---|---|
| sin(90° − θ) | cos θ |
| cos(90° − θ) | sin θ |
| tan(90° − θ) | cot θ |
| cot(90° − θ) | tan θ |
| sec(90° − θ) | cosec θ |
| cosec(90° − θ) | sec θ |
Memory Trick: “co-function” means the function changes to its partner.
6. Signs in Different Quadrants
As angles move beyond 90°, signs change according to the quadrant.
| Quadrant | Positive Functions |
|---|---|
| I (0°–90°) | All |
| II (90°–180°) | Sin, Cosec |
| III (180°–270°) | Tan, Cot |
| IV (270°–360°) | Cos, Sec |
Memory Trick: All Students Take Coffee (ASTC)
7. Pythagoras and Trigonometry
Many trigonometric identities come directly from the Pythagorean theorem.
a² + b² = c²
Where:
- a = Perpendicular
- b = Base
- c = Hypotenuse
If any two sides are known, the third side can be found instantly.
8. Angle Transformation Rules
Useful when angles exceed 90°.
| Expression | Result |
|---|---|
| sin(180° − θ) | sin θ |
| cos(180° − θ) | −cos θ |
| tan(180° − θ) | −tan θ |
| sin(180° + θ) | −sin θ |
| cos(180° + θ) | −cos θ |
| tan(180° + θ) | tan θ |
Questions involving large angles usually become simple after applying these transformations.
9. Product Relationships Worth Memorizing
| Formula |
|---|
| tan θ = sin θ / cos θ |
| cot θ = cos θ / sin θ |
| sec θ = 1 / cos θ |
| cosec θ = 1 / sin θ |
These conversions are often enough to simplify complex expressions without using advanced identities.
10. Common Mistakes to Avoid
- Do not assume tan θ = sin θ × cos θ.
- Never cancel terms across addition signs.
- Remember tan 90° is undefined.
- Always check the quadrant before deciding the sign.
- Convert sec, cosec and cot into sin and cos when stuck.
11. Fast Problem-Solving Strategy
- Write everything in terms of sin and cos.
- Apply the identity sin²θ + cos²θ = 1 wherever possible.
- Use standard angle values.
- Reduce complicated expressions step by step.
- Check signs based on the quadrant.
This approach solves a surprisingly large percentage of trigonometry questions.
12. The Ultimate 30-Second Revision Box
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
- tan θ = sin θ / cos θ
- sin(90° − θ) = cos θ
- cos(90° − θ) = sin θ
- Remember 0°, 30°, 45°, 60°, 90° values
- Use ASTC for signs in quadrants
- Convert reciprocal functions when simplifying
- Most questions become easier after rewriting everything using sine and cosine