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Tamil Nadu (TNBSE)Class 10 Mathematics★ 7-11 marks in the TN SSLC Class 10 Maths public exam through MCQs, identity proofs and heights-and-distances problems

Trigonometry

Samacheer Kalvi (TN State Board) Class 10 Maths Chapter 6, Trigonometry. Trigonometric ratios and identities (the square relations), and heights and distances using angles of elevation and depression. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

⏱ 20 min readMedium difficulty✓ Free · NCERT-aligned

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

1Use the six trigonometric ratios
2Apply the square-relation identities
3Prove simple trigonometric identities
4Solve heights-and-distances problems
5Use angles of elevation and depression

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Why this chapter matters

Trigonometry connects angles with lengths and is used to find heights and distances that cannot be measured directly. Identity proofs and elevation/depression problems are reliable scoring questions in the TN SSLC exam.

Before you start — revise these

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

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Class 9 — Introduction to trigonometry

Trigonometric ratios and standard angles.

Trigonometry — Class 10 Maths (Samacheer Kalvi)

TN State Board (Samacheer Kalvi) Class 10 Mathematics, Chapter 6. Ratios, identities and real-world heights and distances.

1. About this chapter

This chapter covers trigonometric ratios, the trigonometric identities (square relations), and heights and distances using angles of elevation and depression.

2. Trigonometric ratios and identities

For a right triangle: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj, and their reciprocals cosec, sec, cot.
Square-relation identities:
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ

3. Heights and distances

Angle of elevation: the angle above the horizontal when looking up at an object.
Angle of depression: the angle below the horizontal when looking down.
Use tan θ = height / distance (and other ratios) to find unknown heights or distances.
Standard values: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

4. Worked examples

Example 1. Prove sin²θ + cos²θ = 1 is consistent for θ = 30°. sin30° = ½, cos30° = √3/2 → (½)² + (√3/2)² = ¼ + ¾ = 1 ✓.

Example 2. A tower casts the angle of elevation 45° from a point 20 m away. Find its height. tan45° = h/20 → 1 = h/20 → h = 20 m.

Example 3. If tan θ = 3/4, find sec θ. sec²θ = 1 + tan²θ = 1 + 9/16 = 25/16 → sec θ = 5/4.

5. Common mistakes

Mistake: Writing 1 + tan²θ = cosec²θ. Fix: 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
Mistake: Confusing elevation and depression. Fix: Looking up = elevation; looking down = depression (angles are equal alternate angles).
Mistake: Mixing the sides in a ratio. Fix: Always identify opposite, adjacent and hypotenuse relative to θ.

6. Practice (book-back style)

State the three square-relation identities.
If sin θ = 3/5, find cos θ.
A ladder makes 60° with the ground and reaches 6 m up a wall. Find the ladder's length.
Define the angle of depression.
Evaluate tan 60° − tan 30°.

7. Answer key

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25 → cos θ = 4/5.
sin60° = 6/L → √3/2 = 6/L → L = 12/√3 = 4√3 m.
The angle below the horizontal when an observer looks down at an object.
√3 − 1/√3 = (3 − 1)/√3 = 2/√3.

8. Quick revision

Chapter 6 · ratios, identities, heights and distances.
sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
tan θ = height/distance for heights and distances.
Elevation = looking up; depression = looking down.
tan30° = 1/√3, tan45° = 1, tan60° = √3.

Key formulas & results

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

Square-relation identities

sin²θ + cos²θ = 1

Fundamental identity.

Secant identity

1 + tan²θ = sec²θ

Derived identity.

Cosecant identity

1 + cot²θ = cosec²θ

Derived identity.

Heights and distances

tan θ = height / distance

Use elevation/depression angles.

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Common mistakes & fixes

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

WATCH OUT

✗ Writing 1 + tan²θ = cosec²θ

✓ 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

WATCH OUT

✗ Confusing elevation and depression

✓ Looking up = elevation; looking down = depression.

WATCH OUT

✗ Mixing the sides in a ratio

✓ Identify opposite, adjacent and hypotenuse relative to θ.

NCERT exercises (with solutions)

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

Multiple choice questions

MCQs on ratios, identities and standard angles.

Questions

Identity proofs

Prove trigonometric identities using the square relations.

Questions

Heights and distances

Elevation and depression problems.

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Recall

State the three square-relation identities.

▶ Show solution

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Q2EASY· Numerical

If sin θ = 3/5, find cos θ.

▶ Show solution

cos²θ = 1 − 9/25 = 16/25 → cos θ = 4/5.

Q3MEDIUM· Application

A ladder makes 60° with the ground and reaches 6 m up a wall. Find the ladder's length.

▶ Show solution

sin60° = 6/L → √3/2 = 6/L → L = 12/√3 = 4√3 m.

Q4MEDIUM· Numerical

If tan θ = 3/4, find sec θ.

▶ Show solution

sec²θ = 1 + 9/16 = 25/16 → sec θ = 5/4.

Q5EASY· Application

A tower subtends 45° elevation from a point 20 m away. Find its height.

▶ Show solution

tan45° = h/20 → h = 20 m.

Q6EASY· Concept

Define the angle of depression.

▶ Show solution

The angle below the horizontal when an observer looks down at an object.

5-minute revision

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

•Chapter 6 of Samacheer Kalvi Class 10 Mathematics.
•sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
•Heights and distances use tan θ = height/distance.
•Elevation = looking up; depression = looking down.
•tan30° = 1/√3, tan45° = 1, tan60° = √3.
•Draw a labelled right triangle for each problem.

Tamil Nadu (TNBSE) marks blueprint

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

Typical chapter weightage: 7-11 marks across MCQ, proofs and application problems

Question type	Marks each	Typical count	What it tests
MCQ	1	1-2	Ratios and identities
Identity Proof	2-3	1-2	Square-relation identities
Heights & Distances	2-5	1	Elevation/depression problems

Prep strategy

Memorise the three identities and standard angle values
Practise identity proofs
Draw figures for heights-and-distances
Pick the right ratio (sin/cos/tan)

Where this shows up in the real world

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

Surveying

Heights of towers and hills are found using elevation angles.

Navigation and astronomy

Angles and trigonometry locate ships, stars and satellites.

Architecture

Roof slopes and ramps use trigonometric ratios.

Exam strategy

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

Draw and label the right triangle
Choose the ratio that links known and unknown
Use exact standard-angle values
Pick the correct identity for proofs

Going beyond the textbook

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

Prove (1 + cot θ − cosec θ)(1 + tan θ + sec θ) = 2.
Find a height using two angles of elevation from two points.

Where else this chapter is tested

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

TN SSLC Class 10 Public ExamHigh

Foundation / NTSE MathematicsMedium

School unit testsHigh

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Questions students ask

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

They are alternate angles formed by a horizontal line of sight and the line joining the two points, so they are equal.

Replacing one trigonometric expression with an equivalent identity (like sec²θ = 1 + tan²θ) simplifies one side of an equation until it matches the other.

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