Class 10 Mathematics – Chapter 8: Introduction to Trigonometry

1. Introduction to Trigonometry

Trigonometry comes from two Greek words: trigonon (triangle) and metron (measure). It began as the study of triangles — first in astronomy and map-making — and in Class 10 we study its foundation: the link between an acute angle of a right-angled triangle and the ratio of two sides of that triangle.

The key observation: in right-angled triangles of every size, if the acute angle is the same, then the ratios of corresponding sides are the same. A small triangle and a huge triangle that both contain a \(60^\circ\) angle both give the value \(\frac{1}{2}\) for \(\sin 60^\circ\). The ratio is a pure number attached to the angle, with no unit of length attached.

Fix an acute angle θ in a right-angled triangle (draw it at vertex \(A\) of \(\triangle ABC\), right-angled at \(B\)):

• Perpendicular (P) — the side facing the angle, opposite to it: \(BC\).
• Base (B) — the side along which the angle rests, adjacent to it: \(AB\).
• Hypotenuse (H) — the longest side, facing the right angle: \(AC\).

2. Trigonometric Ratios

For a fixed acute angle θ in a right-angled triangle with perpendicular \(P\), base \(B\) and hypotenuse \(H\), we define three main ratios:

Their reciprocals are the other three ratios:

Notice how each reciprocal simply turns the original fraction upside down:

• \(\mathrm{cosec}\,\theta = \frac{1}{\sin \theta}\) — reciprocal of sine.
• \(\sec \theta = \frac{1}{\cos \theta}\) — reciprocal of cosine.
• \(\cot \theta = \frac{1}{\tan \theta}\) — reciprocal of tangent.

Two more useful consequences come straight from the definitions:

• \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \] because \(\frac{P}{H} \div \frac{B}{H} = \frac{P}{B}\).
• \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \] because \(\frac{B}{H} \div \frac{P}{H} = \frac{B}{P}\).

3. Trigonometric Ratios of Specific Angles

We do not need a calculator for the angles \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\): their exact values come from two simple right triangles and the definitions above.

Right triangle with angles \(30^\circ, 60^\circ, 90^\circ\)

Half of an equilateral triangle of side 2 gives a right-angled triangle with sides \(1, \sqrt{3}, 2\): the shortest side \(1\) is opposite the smallest angle \(30^\circ\), the side \(\sqrt{3}\) is opposite \(60^\circ\), and the hypotenuse is \(2\).

Right triangle with angles \(45^\circ, 45^\circ, 90^\circ\)

A square divided along a diagonal gives a right-angled triangle with two equal legs \(1, 1\) and hypotenuse \(\sqrt{1^2 + 1^2} = \sqrt{2}\). Both acute angles are \(45^\circ\).

The standard-value table

One-line memory pattern

For \(\sin \theta\) the values are \(0, \frac{1}{2}, \frac{1}{\sqrt{2}}, \frac{\sqrt{3}}{2}, 1\) reading the columns 0° to 90°; for \(\cos \theta\) the same column reads back to front. \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) every time, and each reciprocal fills its own row.

4. Finding Trigonometric Ratios

Given a right-angled triangle, the four-step routine is: (1) mark the chosen acute angle, (2) write down \(P, B, H\) for that angle, (3) use Pythagoras if only two sides are given, (4) substitute into the required ratio.

5. Trigonometric Identities

The three fundamental identities follow in one line each from Pythagoras' theorem \(P^2 + B^2 = H^2\).

Derivation

Start from Pythagoras: \(P^2 + B^2 = H^2\). This holds for every right triangle.

6. Simplification of Trigonometric Expressions

Class 10 simplification problems combine the reciprocal relationships, the three identities and ordinary algebra. Convert everything to \(\sin\theta\) and \(\cos\theta\) if you get stuck — that route always works.

7. Evaluation of Trigonometric Expressions

To evaluate an expression, replace each standard angle by its exact value from the table and simplify the arithmetic exactly (keep surds, avoid decimals).

8. Finding Unknown Ratios

When one ratio is given, the others come from a right triangle built with the given \(P, B, H\), or directly from the identities. Both routes are equally acceptable in CBSE answers.

9. Application and Competency-Based Questions

These problems test whether you can spot the correct ratio inside a described right-angled situation. The angle stays inside the triangle — nothing is measured across open ground; height-and-distance questions with elevation angles belong to the separate chapter Some Applications of Trigonometry.

10. Important Formulas

11. Common Mistakes in Introduction to Trigonometry

12. CBSE Exam Tips

13. Quick Revision

14. Worked Examples — Consolidated Section

Thirty fully solved examples triangulate the whole chapter: basic concepts (1–6), finding ratios (7–14), standard-angle evaluation (15–20), identity-based simplification (21–26) and application/competency problems (27–30).

15. Self-Practice Questions

Thirty-two questions covering every skill in this chapter. Answer them fully before checking the answer list, which is given separately at the end.

A. Identifying ratios

1. In \(\triangle ABC\), right-angled at \(B\), with the acute angle at \(C\) marked as θ, name the perpendicular, base and hypotenuse.
2. The sides of a right triangle are 6, 8 and 10. Which side is the hypotenuse?
3. Can \(\sin\theta\) be greater than 1? Give a one-line reason.
4. Which ratio equals \(\frac{\text{Base}}{\text{Hypotenuse}}\)? Which equals \(\frac{\text{Perpendicular}}{\text{Base}}\)?

B. Calculating ratios from sides

5. A right triangle has \(P = 9\), \(B = 12\), \(H = 15\). Write \(\sin\theta\), \(\cos\theta\), \(\tan\theta\).
6. A right triangle has \(P = 5\), \(B = 12\), \(H = 13\). Write all six ratios of the angle.
7. For the 8-15-17 triangle, with \(P = 8\) and \(B = 15\), find \(\mathrm{cosec}\,\theta\), \(\sec\theta\), \(\cot\theta\).
8. If \(\sin\theta = \frac{12}{13}\), write the corresponding values of \(P\), \(B\), \(H\) and hence \(\cos\theta\).

C. Finding the missing side

9. In a right triangle, \(B = 12\) and \(H = 20\). Find \(P\), then write \(\tan\theta\).
10. In a right triangle, \(P = 7\) and \(H = 25\). Find \(B\), then write \(\sec\theta\).

D. Standard-angle values

11. Write the values of \(\sin 30^\circ\), \(\cos 60^\circ\) and \(\tan 45^\circ\).
12. Write \(\sin 0^\circ\), \(\cos 90^\circ\), \(\sin 90^\circ\), \(\cos 0^\circ\).
13. Write \(\tan 30^\circ\), \(\tan 60^\circ\) and \(\cot 45^\circ\).
14. Which of \(\tan 90^\circ\) and \(\sin 90^\circ\) is not defined?

E. Reciprocal ratios

15. State the reciprocal of each of \(\sin\theta\), \(\cos\theta\), \(\tan\theta\).
16. If \(\sin\theta = \frac{4}{5}\), find \(\mathrm{cosec}\,\theta\).
17. If \(\sec\theta = \frac{17}{15}\), find \(\cos\theta\) and \(\sin\theta\).

F. Fundamental identities

18. Evaluate \(\sin^2 30^\circ + \cos^2 30^\circ\).
19. If \(\tan\theta = \sqrt3\), use \(1 + \tan^2\theta = \sec^2\theta\) to find \(\sec\theta\).
20. If \(\cot\theta = 1\), use \(1 + \cot^2\theta = \mathrm{cosec}^2\theta\) to find \(\mathrm{cosec}\,\theta\).

G. Simplification

21. Simplify \((1 + \tan^2\theta)\cos^2\theta\).
22. Simplify \(\dfrac{1 - \sin^2\theta}{1 - \cos^2\theta}\).
23. Simplify \(\dfrac{1}{1 + \cot^2\theta}\).
24. Prove that \(\sin\theta \cdot \cot\theta = \cos\theta\).

H. Evaluation of expressions

25. Evaluate \(\sin 45^\circ + \cos 45^\circ\).
26. Evaluate \(\tan 45^\circ + \tan 30^\circ \cdot \tan 60^\circ\).
27. Evaluate \(\sin^2 30^\circ + \cos^2 30^\circ\) from the table (do not use the identity).

I. Finding unknown ratios

28. If \(\cos\theta = \frac{12}{13}\), find \(\sin\theta\) and \(\tan\theta\).
29. If \(\tan\theta = 1\), find \(\sin\theta\) and \(\cos\theta\).

J. Application and competency

30. A ramp rises 3 m straight up over a base of 4 m, with a strongly slanted length of 5 m. For the angle the ramp makes with the ground, write \(\sin\theta\), \(\cos\theta\), \(\tan\theta\).
31. A right triangle has hypotenuse 10 and one perpendicular side 6. Find the remaining side and \(\tan\theta\) for that angle.
32. If \(\sin\theta = \frac{5}{13}\), evaluate \(1 + \tan^2\theta\).

Answers

16. Related Resources

• Introduction to Trigonometry — Practice Paper
• Introduction to Trigonometry — Quiz
• Introduction to Trigonometry — Chapter Test
• Introduction to Trigonometry — Chapter Overview
• Class 10 Mathematics
• Class 10 Study Material
• Chapter 7 — Coordinate Geometry Notes
• Chapter 6 — Triangles Notes