1. Trigonometric Ratios
In right triangle ABC, right-angled at B:
• sin ? = Opposite/Hypotenuse = AB/AC
• cos ? = Adjacent/Hypotenuse = BC/AC
• tan ? = Opposite/Adjacent = AB/BC
• cosec ? = 1/sin ? = AC/AB
• sec ? = 1/cos ? = AC/BC
• cot ? = 1/tan ? = BC/AB
Mnemonic: Some People Have Curly Black Hair Through Proper Brushing
Sin = P/H, Cos = B/H, Tan = P/B
2. Trigonometric Ratios of Specific Angles
| Angle (?) |
sin ? |
cos ? |
tan ? |
cosec ? |
sec ? |
cot ? |
| 0° |
0 |
1 |
0 |
Not defined |
1 |
Not defined |
| 30° |
1/2 |
v3/2 |
1/v3 |
2 |
2/v3 |
v3 |
| 45° |
1/v2 |
1/v2 |
1 |
v2 |
v2 |
1 |
| 60° |
v3/2 |
1/2 |
v3 |
2/v3 |
2 |
1/v3 |
| 90° |
1 |
0 |
Not defined |
1 |
Not defined |
0 |
3. Trigonometric Identities
Pythagorean Identities:
1. sin˛ ? + cos˛ ? = 1
2. 1 + tan˛ ? = sec˛ ?
3. 1 + cot˛ ? = cosec˛ ?
Reciprocal Identities:
• cosec ? = 1/sin ?
• sec ? = 1/cos ?
• cot ? = 1/tan ?
Quotient Identities:
• tan ? = sin ?/cos ?
• cot ? = cos ?/sin ?
4. Complementary Angles
sin(90° - ?) = cos ?
cos(90° - ?) = sin ?
tan(90° - ?) = cot ?
cot(90° - ?) = tan ?
sec(90° - ?) = cosec ?
cosec(90° - ?) = sec ?
Note: 90° - ? and ? are complementary angles
5. Trigonometric Ratios of 0° and 90°
At 0°
sin 0° = 0
cos 0° = 1
tan 0° = 0
At 90°
sin 90° = 1
cos 90° = 0
tan 90° = 8 (undefined)
6. Important Formulas
• sin˛ ? = 1 - cos˛ ?
• cos˛ ? = 1 - sin˛ ?
• tan˛ ? = sec˛ ? - 1
• cot˛ ? = cosec˛ ? - 1
• sin ? = v(1 - cos˛ ?)
• cos ? = v(1 - sin˛ ?)
7. Trigonometric Ratios Table (Simplified)
Remember:
sin 0° = 0/4 = 0
sin 30° = 1/4 = 1/2
sin 45° = 2/4 = 1/v2
sin 60° = 3/4 = v3/2
sin 90° = 4/4 = 1
8. Solving Trigonometric Equations
- Use identities to simplify equation
- Express everything in terms of sin ? and cos ?
- Solve the algebraic equation
- Find angle using trigonometric table
- Check if solution is within valid range (0° to 90°)
9. Quick Reference
Basic Ratios
sin = P/H
cos = B/H
tan = P/B
Key Identity
sin˛? + cos˛? = 1
Complementary
sin(90°-?) = cos ?
10. Example (NCERT Style)
Problem: If sin A = 3/5, find cos A and tan A
Solution:
Using sin˛A + cos˛A = 1
(3/5)˛ + cos˛A = 1
9/25 + cos˛A = 1
cos˛A = 1 - 9/25 = 16/25
cos A = 4/5 (positive in first quadrant)
tan A = sin A/cos A = (3/5)/(4/5) = 3/4
Problem: Evaluate: 2 tan˛ 45° + cos˛ 30° - sin˛ 60°
Solution:
tan 45° = 1, cos 30° = v3/2, sin 60° = v3/2
= 2(1)˛ + (v3/2)˛ - (v3/2)˛
= 2 + 3/4 - 3/4 = 2
11. Special Values Memory Aid
For sin: 0°, 30°, 45°, 60°, 90°
Values: v0/2, v1/2, v2/2, v3/2, v4/2
Which gives: 0, 1/2, 1/v2, v3/2, 1
For cos: Reverse of sin
cos 0° = sin 90° = 1
cos 30° = sin 60° = v3/2
cos 45° = sin 45° = 1/v2
cos 60° = sin 30° = 1/2
cos 90° = sin 0° = 0