Introduction to Trigonometry | Class 10 Mathematics Notes

8.3 Trigonometric Ratios of Some Specific Angles

This section focuses on calculating the exact values of trigonometric ratios for special angles 0°, 30°, 45°, 60°, and 90°, which are frequently used in geometry and trigonometry.

For 45°: Consider a right triangle ABC right angled at B with angles A and C both 45°. Since the angles are equal, the sides opposite these angles are equal, i.e., BC = AB = a. Using Pythagoras theorem, the hypotenuse AC = a√2. Therefore:

• sin 45° = BC / AC = a / (a√2) = 1 / √2
• cos 45° = AB / AC = a / (a√2) = 1 / √2
• tan 45° = BC / AB = a / a = 1

The reciprocals are:

• cosec 45° = √2
• sec 45° = √2
• cot 45° = 1

For 30° and 60°: Consider an equilateral triangle ABC with each side 2a and each angle 60°. Draw the perpendicular AD from A to BC, bisecting BC into BD and DC each of length a. Triangle ABD is right angled at D with angles 30° at A and 60° at B.

Using Pythagoras theorem:

• AD = a√3

Therefore:

• sin 30° = BD / AB = a / 2a = 1/2
• cos 30° = AD / AB = (a√3) / 2a = √3 / 2
• tan 30° = BD / AD = a / (a√3) = 1 / √3

Reciprocals:

• cosec 30° = 2
• sec 30° = 2 / √3
• cot 30° = √3

Similarly for 60°:

• sin 60° = AD / AB = √3 / 2
• cos 60° = BD / AB = 1/2
• tan 60° = AD / BD = √3

Reciprocals:

• cosec 60° = 2 / √3
• sec 60° = 2
• cot 60° = 1 / √3

For 0° and 90°: As angle A approaches 0°, side BC approaches 0 and AB approaches AC, so sin 0° = 0, cos 0° = 1. As angle A approaches 90°, side AB approaches 0 and BC approaches AC, so sin 90° = 1, cos 90° = 0.

Other ratios for 0° and 90° are undefined where division by zero occurs.

Table 8.1 summarizes all these values for quick reference.

These exact values are crucial for solving trigonometric problems efficiently without a calculator.

📊 Diagram: Fig. 8.14 shows right triangle ABC right angled at B with angles 45° each; Fig. 8.15 shows equilateral triangle ABC with perpendicular AD drawn to BC, forming two right triangles with angles 30° and 60°; Fig. 8.16 and Fig. 8.17 illustrate the limiting behavior of sides as angle A approaches 0° and 90° respectively.

🧪 Activity: Students are encouraged to construct these special triangles and verify the trigonometric ratios experimentally.

🔗 Connection: Sets the foundation for solving problems using these standard values and understanding trigonometric identities in section 8.4.

Table on page 13 (7×6)