Chapter 8 of 14

57%

Odisha (BSE)Class 10 Mathematics★ Unit V — Trigonometry: 12 marks, shared with Some Applications of Trigonometry · CBSE Class 10 Maths (Chapter 8)

Introduction to Trigonometry

Introduction to Trigonometry — sin, cos, tan ratios in right triangle. Trigonometric ratios of specific angles (0°, 30°, 45°, 60°, 90°). Trigonometric identities. CBSE Class 10 Maths Chapter 8.

⏱ 26 min readMedium difficulty✓ Free · NCERT-aligned

Worksheet

Podcast

Listen as a conversation

A teacher & a student talk through the chapter — in your language.

Mind Map

The chapter at a glance

Every concept and exam-favourite branch — revisable in 5 minutes.

…

Study Guide

Night-before notes

Key points, glossary and the questions most likely to show up.

…

VocabularyBuild your vocabularyLexiGenome mines the key words and schedules them with spaced repetition.Mine words →

By the end of this chapter you'll be able to…

1Define the six trigonometric ratios from the sides of a right triangle
2Given one ratio, reconstruct the triangle and find all the others
3Recall the ratios at 0°, 30°, 45°, 60° and 90°, and know which are undefined
4Use the three identities to prove further identities
5Judge which statements about trigonometric ratios are true, and justify the verdict

💡

Why this chapter matters

Trigonometry turns an angle into a number you can compute with. Everything in Chapter 9 — heights, distances, angles of elevation — runs on the ratios defined here, and so does most of Class 11 and 12 mathematics and all of wave physics.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

Pythagoras foundation

Introduction to Trigonometry — Class 10 Mathematics

What CBSE examines here (2026-27). The six trigonometric ratios and how to get all of them from any one; the ratios of 0°, 30°, 45°, 60° and 90°; and the three trigonometric identities. The chapter runs to three exercises — 8.1, 8.2 and 8.3. Trigonometric ratios of complementary angles were removed from this chapter; they are in the appendix at the end, marked as background.

"Triangles + Ratios = Trigonometry. The mathematics that measures the sky."

1. About the Chapter

Trigonometry = 'measurement of triangles' (Greek: 'trigonon' + 'metron'). Used in:

Astronomy (measuring stars)
Surveying (measuring heights)
Engineering (forces, designs)
Physics (waves, oscillations)

This chapter introduces the six trigonometric ratios and basic identities.

Indian Heritage

Aryabhata (5th century CE): defined sine ('jya'); coined trigonometric ratios
Brahmagupta, Bhaskara II: refined and extended trigonometry
Sanskrit 'jya' became Arabic 'jiba' became Latin 'sinus' → English 'sine'

2. Right-Angled Triangle Review

For △ABC with right angle at B:

         A
         /|
        / |
       /  | ← perpendicular (P)
      /θ_|
     B    C
     base (B)
   hypotenuse from B to A

Wait, let me redraw. In a right triangle with right angle at C:

Side opposite to angle A = a (perpendicular for angle A)
Side opposite to angle B = b
Side opposite to right angle (C) = c (hypotenuse)

For ANGLE θ (one of the acute angles):

Opposite side: side opposite to θ
Adjacent side: side next to θ (not hypotenuse)
Hypotenuse: longest side, opposite to 90° angle

3. The Six Trigonometric Ratios

For an acute angle θ in a right triangle:

Sine (sin θ)

sin θ = Opposite / Hypotenuse = P / H

Cosine (cos θ)

cos θ = Adjacent / Hypotenuse = B / H

Tangent (tan θ)

tan θ = Opposite / Adjacent = P / B

Cosecant (cosec θ) — reciprocal of sin

cosec θ = Hypotenuse / Opposite = H / P = 1 / sin θ

Secant (sec θ) — reciprocal of cos

sec θ = Hypotenuse / Adjacent = H / B = 1 / cos θ

Cotangent (cot θ) — reciprocal of tan

cot θ = Adjacent / Opposite = B / P = 1 / tan θ

Memory Aid (SOH CAH TOA)

SOH: Sin = Opposite/Hypotenuse
CAH: Cos = Adjacent/Hypotenuse
TOA: Tan = Opposite/Adjacent

4. Reciprocal Relations

These give 3 more ratios:

sin θ × cosec θ = 1 → cosec θ = 1/sin θ
cos θ × sec θ = 1 → sec θ = 1/cos θ
tan θ × cot θ = 1 → cot θ = 1/tan θ

Also:

tan θ = sin θ / cos θ
cot θ = cos θ / sin θ

5. Trigonometric Ratios of Specific Angles

TABLE (MEMORISE!)

Angle	sin	cos	tan	cosec	sec	cot
0°	0	1	0	∞	1	∞
30°	1/2	√3/2	1/√3	2	2/√3	√3
45°	1/√2	1/√2	1	√2	√2	1
60°	√3/2	1/2	√3	2/√3	2	1/√3
90°	1	0	∞	1	∞	0

Memory Tricks

sin 0°, 30°, 45°, 60°, 90° = √0/2, √1/2, √2/2, √3/2, √4/2
That is: 0, 1/2, √2/2, √3/2, 1

cos is sin in REVERSE order:

cos 0°, 30°, 45°, 60°, 90° = 1, √3/2, √2/2, 1/2, 0

tan = sin / cos (gets ∞ where cos = 0)

6. Pythagoras and Trigonometry

From Pythagoras: P² + B² = H²

Divide by H²: (P/H)² + (B/H)² = 1

This gives: sin²θ + cos²θ = 1 — fundamental identity!

7. Trigonometric Identities (MEMORISE)

Three Key Identities

Identity 1: sin²θ + cos²θ = 1

Identity 2: 1 + tan²θ = sec²θ

Identity 3: 1 + cot²θ = cosec²θ

Derivations

Identity 2: Divide Identity 1 by cos²θ:

sin²θ/cos²θ + 1 = 1/cos²θ
tan²θ + 1 = sec²θ → 1 + tan²θ = sec²θ ✓

Identity 3: Divide Identity 1 by sin²θ:

1 + cos²θ/sin²θ = 1/sin²θ
1 + cot²θ = cosec²θ ✓

Useful Forms

sin²θ = 1 − cos²θ
cos²θ = 1 − sin²θ
sec²θ − tan²θ = 1
cosec²θ − cot²θ = 1

8. Worked Examples

Example 1: Find Ratios

In right △ABC, ∠B = 90°, AB = 3, BC = 4. Find sin C, cos C, tan C.

AC (hypotenuse) = √(9+16) = 5
For angle C: opposite = AB = 3, adjacent = BC = 4, hypotenuse = 5
sin C = 3/5, cos C = 4/5, tan C = 3/4

Example 2: Use Identity

Find sin A if cos A = 5/13.

sin²A + cos²A = 1
sin²A = 1 − 25/169 = 144/169
sin A = 12/13

Example 3: Specific Values

Evaluate: sin 30° × cos 60° + sin 60° × cos 30°

= (1/2)(1/2) + (√3/2)(√3/2)
= 1/4 + 3/4 = 1

(This is actually sin(30° + 60°) = sin 90° = 1, using formula sin(A+B), but for Class 10 we calculate directly.)

Example 4: Prove Identity

Prove: (1 − sin²θ) × sec²θ = 1

LHS = cos²θ × (1/cos²θ) = 1
= RHS ✓

Example 5: One Ratio to All the Others

Given cot θ = 7/8, find (1 + sin θ)(1 − sin θ) / [(1 + cos θ)(1 − cos θ)].

Each bracket pair is a difference of squares: the numerator is 1 − sin²θ and the denominator is 1 − cos²θ.
By the first identity those are cos²θ and sin²θ, so the whole expression is cos²θ/sin²θ = cot²θ.
cot θ = 7/8, so the value is 49/64.

Recognising the difference of squares turns a four-bracket expression into a one-line answer.

9. Common Mistakes

Confusing opposite and adjacent
- 'Opposite' is OPPOSITE to the angle θ; 'Adjacent' is the OTHER side (not hypotenuse).
Wrong specific angle values
- MEMORISE the table. Don't guess.
Forgetting reciprocal relations
- cosec ≠ cos. cosec = 1/sin.
Identity confusion
- sin²θ + cos²θ = 1 (NOT sin θ + cos θ = 1).
tan at 90°
- tan 90° is UNDEFINED (∞), not 0.

10. Indian Heritage of Trigonometry

Aryabhata (476-550 CE)

Defined SINE (jya), COSINE (kojya), VERSINE
Tables of sines for various angles
Used in astronomy

Brahmagupta (598-668 CE)

Extended Aryabhata's work
Brahmagupta-Fibonacci identity

Bhaskara II (1114-1185 CE)

Refined trigonometric calculations
Indian astronomy depended heavily on this

Madhava (1340-1425 CE)

INFINITE SERIES for sin, cos, arctan
200 years before Newton!
Astonishing achievement

Word Origin

Sanskrit 'jya' → Arabic 'jiba' → Latin 'sinus' → English 'sine'

A direct line from Indian mathematics to global use.

11. Conclusion

Trigonometry is one of the MOST USED branches of mathematics:

Astronomy: distances to stars, planet orbits
Engineering: forces, structures, machines
Physics: waves, oscillations, optics
Computer graphics: rotations, animations

Master:

6 trigonometric ratios (SOH CAH TOA), and the reciprocal pairings
Specific angle values (0°, 30°, 45°, 60°, 90°) — including which ones are undefined
3 identities (sin² + cos² = 1, sec² − tan² = 1, cosec² − cot² = 1)
The habit of rebuilding the right triangle whenever a single ratio is given

Chapter 9 will apply these to REAL-WORLD problems (heights and distances).

Trigonometry: the mathematics of triangles, taught by Indians to the world.

Appendix — beyond the current syllabus

Not examinable in CBSE 2026-27. Trigonometric ratios of complementary angles were removed from this chapter during rationalisation, along with the exercise that went with them. The rationalised chapter runs 8.1 Introduction → 8.2 Trigonometric Ratios → 8.3 Ratios of Some Specific Angles → 8.4 Trigonometric Identities → 8.5 Summary, and the summary's six points say nothing about complementary angles. They are kept here because they are genuinely useful, they explain the names of half the ratios, and every guidebook still teaches them.

The complementary-angle relations

Two angles are complementary when they add to 90°. In a right triangle the two acute angles are always complementary, since the third angle has already used up 90° of the 180°.

That single fact produces the whole set. What is opposite one acute angle is adjacent to the other, so each ratio turns into its "co-" partner:

sin(90° − A) = cos A and cos(90° − A) = sin A
tan(90° − A) = cot A and cot(90° − A) = tan A
sec(90° − A) = cosec A and cosec(90° − A) = sec A

Check them against the table you already know:

sin 30° = cos 60° = 1/2 ✓
tan 45° = cot 45° = 1 ✓ (45° is its own complement)
sec 60° = cosec 30° = 2 ✓

This is where the names come from. Cosine is the sine of the complementary angle — Aryabhata's term kotijya meant exactly that. The same goes for cotangent and cosecant. So the "co-" prefix is not decoration; it records this relationship.

Even though the relations are no longer examined here, the underlying fact — that the two acute angles of a right triangle add to 90° — is used constantly in Chapter 9, where you routinely work out the second angle of a triangle from the first.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

sin θ

opposite / hypotenuse

SOH

cos θ

adjacent / hypotenuse

CAH

tan θ

opposite / adjacent = sin θ / cos θ

TOA

cosec θ

1 / sin θ

Reciprocal of sine — note it pairs with sin, not cos

sec θ

1 / cos θ

Reciprocal of cosine

cot θ

1 / tan θ = cos θ / sin θ

Reciprocal of tangent

Identity 1

sin²θ + cos²θ = 1

Holds for 0° ≤ θ ≤ 90°. Everything else follows from this one

Identity 2

sec²θ − tan²θ = 1, i.e. 1 + tan²θ = sec²θ

Holds for 0° ≤ θ < 90° — tan and sec fail at 90°

Identity 3

cosec²θ − cot²θ = 1, i.e. 1 + cot²θ = cosec²θ

Holds for 0° < θ ≤ 90° — cosec and cot fail at 0°

Range check

sin θ ≤ 1 and cos θ ≤ 1; sec θ ≥ 1 and cosec θ ≥ 1

A quick sanity test on any answer you produce

⚠️

Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Pairing sec with sin and cosec with cos

✓ The crossover is deliberate: sec is the reciprocal of cos, and cosec is the reciprocal of sin. The 'co' in cosec belongs to the name, not to cosine.

WATCH OUT

✗ Writing sin²θ as sin θ²

✓ sin²θ means (sin θ)² — the ratio, squared. sin(θ²) would be the sine of a squared angle, which is something else entirely.

WATCH OUT

✗ Treating sin A as a product of sin and A

✓ 'sin' is the name of an operation, not a quantity. It cannot be cancelled, and sin A ÷ A is not sin.

WATCH OUT

✗ Ignoring where an identity fails

✓ sec²θ − tan²θ = 1 breaks at θ = 90°, and cosec²θ − cot²θ = 1 breaks at θ = 0°, because the ratios involved are undefined there. The textbook states the ranges explicitly.

WATCH OUT

✗ Accepting an impossible value

✓ sin θ and cos θ never exceed 1, and sec θ and cosec θ are never less than 1. A sine of 4/3 is a signal that something upstream went wrong.

WATCH OUT

✗ Working on both sides of an identity at once

✓ Start from one side and transform it into the other. Manipulating both sides together quietly assumes the very thing you are proving.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 8.1

Exercise 8.1

Trigonometric ratios in a right triangle — finding all six from one, and true/false reasoning

Questions

Ex 8.2

Exercise 8.2

Ratios of 0°, 30°, 45°, 60° and 90° — evaluation, multiple choice, and true/false

Questions

Ex 8.3

Exercise 8.3

Trigonometric identities — expressing ratios in terms of one another, MCQs, and ten identities to prove

Questions

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Introduction to Trigonometry?

4 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

4 questions~3 min worth ~12 marks in Odisha (BSE) exams

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•The six ratios are defined here only for the acute angles of a right triangle
•sin ↔ cosec, cos ↔ sec, tan ↔ cot are the three reciprocal pairs
•Know the 0°/30°/45°/60°/90° table cold — including that tan 90°, sec 90°, cot 0° and cosec 0° are undefined
•sin increases and cos decreases as the angle grows from 0° to 90°
•Three identities: sin²+cos²=1, sec²−tan²=1, cosec²−cot²=1
•Given one ratio, draw the right triangle and get the third side from Pythagoras — then read everything off
•Trigonometric ratios of complementary angles were removed from this chapter
•Aryabhata called the sine jya and the cosine kotijya; 'sine' reached Europe through a mistranslation of the Arabic jaib

Odisha (BSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 10-12 marks

Question type	Marks each	Typical count	What it tests
MCQ	1	3	Definitions, specific values
Short Answer	2-3	2	Identities, ratios
Long Answer	5	1	Prove identities, find all ratios

Prep strategy

MEMORISE table of specific angles cold
MEMORISE 3 identities
Practice 20+ proof problems
Master SOH CAH TOA mnemonic

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Surveying

Indian Survey department uses trigonometry to map terrain.

Astronomy

Aryabhata used trigonometry for astronomical calculations 1500 years ago.

Engineering

Bridge angles, roof slopes, machine parts all use trigonometry.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

When one ratio is given, sketch the right triangle and label two sides — Pythagoras supplies the third

For identity proofs, start from the messier side; if you stall, convert everything to sin and cos

In true/false questions the justification carries the marks, not the verdict

Check every ratio you produce against the range limits before writing it down

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

STRETCH

sin(A+B), cos(A+B) formulas (Class 11)

STRETCH

Trigonometric equations

STRETCH

Inverse trigonometric functions

STRETCH

Hyperbolic functions

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 10 BoardVery High

Maths OlympiadVery High

JEEVery High

NEET PhysicsVery High

AI helpers for this chapter

Generate more practice on the fly, or have the chapter explained at a different level.

⚡

Generate 5 more questions

Fresh MCQs + short-answer items based on this chapter, plus answer keys.

🧠

Explain at a different level

Re-explain the chapter at the level you actually want.

📝 My notes

Saved on this device — sign in to sync · how this works

0/600

Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

No. Trigonometric ratios of complementary angles were removed from this chapter when the syllabus was rationalised, along with their exercise. The rationalised chapter runs Introduction → Trigonometric Ratios → Ratios of Some Specific Angles → Trigonometric Identities → Summary, and the summary lists six points, none about complementary angles. The relations are on the chapter page in the appendix, marked non-examinable, because they are genuinely useful and every guidebook still carries them.

Three: Exercise 8.1 (11 questions), Exercise 8.2 (4 questions) and Exercise 8.3 (4 questions). The older edition had four, one of them devoted to complementary angles — which is why many websites still show that structure.

Because they would need a division by zero. At 0° the side opposite the angle has shrunk to nothing, so cot 0° = adjacent/opposite and cosec 0° = hypotenuse/opposite both have zero on the bottom. The same happens to tan 90° and sec 90°. 'Not defined' is the correct phrase — writing ∞ loses the mark.

Effectively yes, since Exercise 8.2 is unusable without it. But you can rebuild it if you blank: 45° comes from an isosceles right triangle with legs 1 and hypotenuse √2, and 30° and 60° come from bisecting an equilateral triangle of side 2, which gives sides 1, √3 and 2. The textbook derives them in exactly that way.

Practice more, ask doubts, find a tutor

Odisha (BSE) Class 10 mock tests

Odisha (BSE) Class 10 PYQs

Ask in Q&A

Mathematics flashcards

Related chapters

Students who read Introduction to Trigonometry usually read these next.

Real Numbers

The Fundamental Theorem of Arithmetic, HCF and LCM through prime factorisation, and proofs that numbers like √5 are irrational.

Polynomials

Zeroes of a polynomial, what they mean on a graph, and the relationship between the zeroes and coefficients of a quadratic.

Pair of Linear Equations in Two Variables

Solving a pair of linear equations graphically and algebraically — by substitution and by elimination — plus consistency conditions and word problems. CBSE Class 10 Maths Chapter 3.

Quadratic Equations

Degree-2 equations ax² + bx + c = 0 — solving by factorisation and by the quadratic formula, the discriminant and the nature of roots, and word problems on area, speed, age and geometry. CBSE Class 10 Maths Chapter 4.

🧑‍🏫