Trigonometry — Class 9 Maths (Samacheer Kalvi)
TN State Board (Samacheer Kalvi) Class 9 Mathematics, Chapter 6. Ratios of angles in a right triangle.
1. About this chapter
This chapter introduces the trigonometric ratios, the ratios of specific angles, and the ratios of complementary angles.
2. Trigonometric ratios
For a right triangle with angle θ:
- sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
- Reciprocals: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ; also tan θ = sin θ / cos θ.
3. Ratios of standard angles
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | ½ | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | ½ | 0 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ |
4. Complementary angles
- sin(90° − θ) = cos θ; cos(90° − θ) = sin θ.
- tan(90° − θ) = cot θ; sec(90° − θ) = cosec θ (and conversely).
5. Worked examples
Example 1. Evaluate sin 30° + cos 60°. = ½ + ½ = 1.
Example 2. Show that sin 60° = cos 30°. sin 60° = √3/2 and cos 30° = √3/2 → equal (complementary angles).
Example 3. If tan θ = 1, find θ (0° ≤ θ ≤ 90°). tan 45° = 1 → θ = 45°.
6. Common mistakes
- Mistake: Swapping sin and cos in the standard table. Fix: sin increases 0 → 1; cos decreases 1 → 0 as θ goes 0° → 90°.
- Mistake: Writing sin(90° − θ) = sin θ. Fix: sin(90° − θ) = cos θ.
- Mistake: Mixing the sides of a ratio. Fix: Identify opposite, adjacent and hypotenuse relative to θ.
7. Practice (book-back style)
- Write the values of sin 45° and cos 45°.
- Evaluate tan 30° × tan 60°.
- Simplify sin(90° − θ) − cos θ.
- If cos θ = ½, find θ.
- Evaluate cos² 30° + sin² 30°.
8. Answer key
- sin 45° = 1/√2, cos 45° = 1/√2.
- (1/√3)(√3) = 1.
- sin(90° − θ) = cos θ, so cos θ − cos θ = 0.
- cos 60° = ½ → θ = 60°.
- (√3/2)² + (½)² = ¾ + ¼ = 1.
9. Quick revision
- Chapter 6 · ratios, standard angles, complementary angles.
- sin = opp/hyp, cos = adj/hyp, tan = opp/adj; tan θ = sin θ/cos θ.
- Standard angles 0°, 30°, 45°, 60°, 90° (learn the table).
- sin(90° − θ) = cos θ; tan(90° − θ) = cot θ.
- sin² θ + cos² θ = 1 (check with standard angles).
