Trigonometry is the study of relationships between the sides and angles of a right-angled triangle. The word "trigonometry" comes from Greek words meaning "triangle measurement." In Class 10, we focus on trigonometric ratios of acute angles (angles between 0° and 90°).
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Key Concepts and Definitions
Consider a right-angled triangle ABC where angle B = 90° and angle A = theta (an acute angle).
- Hypotenuse (H): The side opposite to the right angle (AC).
- Perpendicular / Opposite (P): The side opposite to angle A (BC).
- Base / Adjacent (B): The side adjacent to angle A (AB).
The Six Trigonometric Ratios:
Key formulas
Standard Angle Values Table:
| Angle | 0° | 30° | 45° | 60° | 90° |
|----------|-----|---------|---------|---------|-----|
| sin | 0 | 1/2 | 1/sqrt2 | sqrt3/2 | 1 |
| cos | 1 | sqrt3/2 | 1/sqrt2 | 1/2 | 0 |
| tan | 0 | 1/sqrt3 | 1 | sqrt3 | undef |
Key Identities (Pythagorean):
Key formulas
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Worked Examples
In a right triangle, the side opposite to an angle A is 4 cm and the hypotenuse is 5 cm. Find sin A, cos A and tan A.
- sin A = 4/5
- Adjacent side = √(52 - 42) = √(25 - 16) = √(9) = 3
- cos A = 3/5
- tan A = 4/3
Find the value of sin 30° + cos 60°.
- sin 30° = 1/2, cos 60° = 1/2
- Answer = 1/2 + 1/2 = 1
Evaluate: 2 tan2 45° + cos2 30° - sin2 60°
- tan 45° = 1, so 2 × 12 = 2
- cos 30° = sqrt3/2, so cos2 30° = 3/4
- sin 60° = sqrt3/2, so sin2 60° = 3/4
- Answer = 2 + 3/4 - 3/4 = 2
If sin A = 3/5, find cos A and tan A (A is acute).
- cos A = √(1 - sin2 A) = √(1 - 9/25) = √(16/25) = 4/5
- tan A = sin A / cos A = (3/5) / (4/5) = 3/4
Prove that (sin theta / (1 - cos theta)) = cosec theta + cot theta.
- LHS = sin theta / (1 - cos theta). Multiply numerator and denominator by (1 + cos theta):
- = sin theta × (1 + cos theta) / (1 - cos2 theta) = sin theta(1 + cos theta) / sin2 theta
- = (1 + cos theta) / sin theta = 1/sin theta + cos theta/sin theta = cosec theta + cot theta = RHS
If tan theta = 1/sqrt3, find all other trigonometric ratios.
- tan theta = 1/sqrt3 means theta = 30°
- sin 30° = 1/2, cos 30° = sqrt3/2, cosec 30° = 2, sec 30° = 2/sqrt3, cot 30° = sqrt3
Simplify: (1 - sin2 A) × sec2 A
- 1 - sin2 A = cos2 A
- cos2 A × sec2 A = cos2 A × 1/cos2 A = 1
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Common mistakes
> Common mistakes: Students often confuse sin/cos/tan with cosec/sec/cot. Remember cosec is the reciprocal of sin, not cos. Also, tan 90° is undefined — never write it as a finite value. When using identities, always check that the angle is valid (not 0° or 90° for tan/cot).
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Summary
Trigonometric ratios link the angles and sides of a right triangle. The three main ratios are sin, cos, and tan. The three Pythagorean identities are essential for simplification and proofs. Memorise the standard angle values for 0°, 30°, 45°, 60°, and 90° as they are used in almost every problem.