Introduction to Trigonometry
In one line
Trigonometry links the angles of a right-angled triangle to the ratios of its sides.
The six ratios
| Ratio | Means | Reciprocal |
|---|---|---|
| sin θ | Opposite ÷ Hypotenuse | cosec θ = 1/sin θ |
| cos θ | Adjacent ÷ Hypotenuse | sec θ = 1/cos θ |
| tan θ | Opposite ÷ Adjacent = sin θ / cos θ | cot θ = 1/tan θ |
Memory trick: SOH CAH TOASin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
Values to learn
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | not defined |
Shortcut: sin goes √0/2, √1/2, √2/2, √3/2, √4/2 for 0° to 90°. cos is the same list backwards.
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The three identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Complementary angles: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ.
Solved examples
1. Find the other ratiosIf sin A = 3/5, then opposite = 3 and hypotenuse = 5, so adjacent = √(25 − 9) = 4.
cos A = 4/5, tan A = 3/4.
cos A = 4/5, tan A = 3/4.
2. Evaluate2 sin 30° + tan 45° = 2 × 1/2 + 1 = 2.
3. Prove(1 − cos²A) × cosec²A = sin²A × (1/sin²A) = 1.
Common mistakes
- Swapping opposite and adjacent. They change depending on which angle you use.
- Thinking sin²θ means sin(θ²). It means (sin θ)².
- Writing tan 90° = 0. It is not defined, because cos 90° = 0.
Quick check
Revise in 30 seconds
| Ratios | sin = O/H · cos = A/H · tan = O/A |
|---|---|
| Reciprocals | cosec = 1/sin · sec = 1/cos · cot = 1/tan |
| Identities | sin² + cos² = 1 · 1 + tan² = sec² · 1 + cot² = cosec² |
| Key values | sin 30 = 1/2 · sin 45 = 1/√2 · sin 60 = √3/2 · tan 45 = 1 |
| Complements | sin(90 − θ) = cos θ |