Trigonometry
TRIGONOMETRY
Introduction to Trigonometry
| Term | Meaning |
|---|---|
| Hypotenuse | Longest side opposite to right angle |
| Opposite Side | Side opposite to the given angle |
| Adjacent Side | Side next to the given angle |
Right triangle → One angle = 90° → Basis of all trigonometric ratios.
Always identify the angle first, then name the opposite and adjacent sides.
Core Trigonometric Ratios
| Ratio | Formula |
|---|---|
| sin θ | Opposite / Hypotenuse |
| cos θ | Adjacent / Hypotenuse |
| tan θ | Opposite / Adjacent |
| cosec θ | Hypotenuse / Opposite |
| sec θ | Hypotenuse / Adjacent |
| cot θ | Adjacent / Opposite |
sin, cos, tan → Primary ratios
cosec, sec, cot → Reciprocal ratios
cosec, sec, cot → Reciprocal ratios
If sin θ is known, its reciprocal is cosec θ = 1 / sin θ.
Trigonometric Ratios of Standard Angles
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | Not defined |
Angles 30°, 45°, 60° → Most important angles for exam problems.
Remember: tan 90° is not defined because division by zero is not possible.
Complementary Angle Relationships
| Relation | Identity |
|---|---|
| sin(90° − θ) | = cos θ |
| cos(90° − θ) | = sin θ |
| tan(90° − θ) | = cot θ |
| sec(90° − θ) | = cosec θ |
Complementary angles sum = 90° → Exchange sin ↔ cos, tan ↔ cot.
If question involves 90° − θ, always convert using complementary identities.
Trigonometric Identities
| Identity | Formula |
|---|---|
| Identity 1 | sin²θ + cos²θ = 1 |
| Identity 2 | 1 + tan²θ = sec²θ |
| Identity 3 | 1 + cot²θ = cosec²θ |
All identities are derived from sin²θ + cos²θ = 1.
Convert everything into sin and cos when simplifying complex expressions.
Applications: Heights and Distances
| Term | Meaning |
|---|---|
| Angle of Elevation | Angle between horizontal and upward line of sight |
| Angle of Depression | Angle between horizontal and downward line of sight |
Angle of elevation = Angle of depression (alternate interior angles).
Always draw a rough diagram before solving heights and distances problems.