Introduction to Trigonometry
Trigonometry and ratios in a right angle triangle.
Introduction to Trigonometry
Understand the six trigonometric ratios, how they are defined from a right-angled triangle, and how to apply them — step by step, with worked examples.
What is Trigonometry?
The word Trigonometry comes from three ancient Greek words — and understanding that etymology immediately tells you what the subject is about:
This chapter is part of the Class 10 Mathematics syllabus for CBSE, Telangana, and Andhra Pradesh boards. Mastering these ratios forms the foundation for topics like heights and distances, and advanced calculus later on.
Parts of a Right-Angled Triangle
Before defining any ratio, you need to clearly identify the three sides of a right-angled triangle relative to a chosen angle. Consider △ABC with the right angle at B:
With respect to ∠A:
- Hypotenuse (AC) — the side opposite the right angle; always the longest side.
- Opposite side (BC) — the side directly across from ∠A.
- Adjacent side (AB) — the side next to ∠A (not the hypotenuse).
Activity: Why Ratios Stay Constant in Similar Triangles
This is the key insight that gives birth to trigonometric ratios. Take four right-angled triangles that all share angle A but have different sizes. Measure the sides and compute the ratios:
| Triangle | Hypotenuse | Opposite (to ∠A) | Adjacent (to ∠A) | Opp ÷ Hyp | Adj ÷ Hyp |
|---|---|---|---|---|---|
| △ABP | 5 | 4 | 3 | 4/5 | 3/5 |
| △ACQ | 10 | 8 | 6 | 8/10 = 4/5 | 6/10 = 3/5 |
| △ADR | 15 | 12 | 9 | 12/15 = 4/5 | 9/15 = 3/5 |
| △AES | 20 | 16 | 12 | 16/20 = 4/5 | 12/20 = 3/5 |
The Six Trigonometric Ratios
For a right-angled triangle ABC (right angle at B), the six trigonometric ratios for angle A are defined as follows:
| Ratio | Symbol | Definition | In △ABC |
|---|---|---|---|
| Sine | sin A | Opposite ÷ Hypotenuse | BC / AC |
| Cosine | cos A | Adjacent ÷ Hypotenuse | AB / AC |
| Tangent | tan A | Opposite ÷ Adjacent | BC / AB |
| Cosecant | cosec A | Hypotenuse ÷ Opposite | AC / BC |
| Secant | sec A | Hypotenuse ÷ Adjacent | AC / AB |
| Cotangent | cot A | Adjacent ÷ Opposite | AB / BC |
Quick Memory Reference:
Also: sin A / cos A = tan A and cos A / sin A = cot A
Worked Examples – Step by Step
Important Concept Questions
These questions frequently appear in board exams (CBSE, Telangana, AP) as short-answer or reasoning-type questions:
No. Since sin x = opposite ÷ hypotenuse, and the hypotenuse is always the longest side of a right triangle, sin x is always a proper fraction (less than 1). The value 4/3 is greater than 1 — so sin x = 4/3 is impossible.
Both sin A and cos A are defined as a ratio where the denominator is the hypotenuse — the longest side. The numerator (opposite or adjacent) is always shorter than the hypotenuse. Therefore the ratio is always a proper fraction, i.e. strictly between 0 and 1.
No! In trigonometry, sin, cos, tan etc. are not separate numbers to be multiplied. They are function names — "tan A" means "the tangent of angle A", which equals Opposite ÷ Adjacent. Writing tan × A is meaningless.
Yes! Dividing sin A (= Opp/Hyp) by cos A (= Adj/Hyp), the Hyp cancels out and you get Opp/Adj = tan A. Similarly, cos A / sin A = cot A.
Common Mistakes to Avoid
- Swapping opposite and adjacent — always identify the angle first, then label the sides relative to that angle.
- Forgetting to find the third side — most problems require using the Pythagoras theorem before you can compute any ratio.
- Thinking sin A > 1 is possible — it is not; sin A and cos A are always between 0 and 1 for acute angles.
- Treating tan as a standalone number — tan A is a single entity representing a ratio, not tan × A.
- Mixing up reciprocal pairs — cosec is the reciprocal of sin, NOT cos. Remember: co-sine ↔ sec, sine ↔ cosec.
What This Lesson Prepares You For
The concepts introduced here — the six ratios and how to compute them from a right-angled triangle — are the building blocks for every exercise in this chapter and beyond. Once you are comfortable with this introduction, you are ready for:
- Exercise 11.1 — Finding trigonometric ratios for given triangles and angles, and verifying reciprocal relationships.
- Trigonometric Identities — Using sin²A + cos²A = 1 and related identities to simplify and prove expressions.
- Trigonometric Ratios of Specific Angles — Values of sin 30°, cos 45°, tan 60°, and others that appear repeatedly in board exams.
- Applications in Heights and Distances, where these ratios are used to find heights of buildings, towers, and distances across rivers.
- Strengthening your foundation in Similar Triangles, which explains why trigonometric ratios remain constant regardless of triangle size.