Introduction to Trigonometry Class 10
Download the PDF containing Introduction to Trigonometry Class 10 notes and a mind map. Revise the six trigonometric ratios, standard-angle values and identities with clear explanations, worked examples and student FAQs.
Trigonometry connects the angles of a right-angled triangle with ratios of its sides. Once you recognise the opposite side, adjacent side and hypotenuse, you can calculate unknown ratios and simplify expressions. Class 10 Introduction to Trigonometry also develops the algebraic skills needed to prove identities.
These Introduction to Trigonometry Class 10 notes support your 2026 revision with a formula sheet, a trigonometric values table and step-by-step examples. Read the explanations first, use the mind map for recall, and then practise questions independently. Understanding which identity to apply is as important as remembering it.
Understanding the Sides of a Right Triangle
Choose an acute reference angle θ before naming the sides. The hypotenuse is opposite the right angle. The opposite side is opposite θ, and the adjacent side is the side next to θ other than the hypotenuse.
The Reference Angle Matters
If you switch to the other acute angle, the opposite and adjacent sides exchange roles. The hypotenuse stays the same. Always mark the given angle before writing a ratio.
Introduction to Trigonometry Class 10 Formula Sheet
For an acute angle θ in a right triangle, let O represent the opposite side, A the adjacent side and H the hypotenuse.
| Ratio | Definition | Related Expression |
|---|---|---|
| sin θ | O/H | 1/cosec θ |
| cos θ | A/H | 1/sec θ |
| tan θ | O/A | sin θ/cos θ |
| cosec θ | H/O | 1/sin θ |
| sec θ | H/A | 1/cos θ |
| cot θ | A/O | cos θ/sin θ |
The memory aid SOH–CAH–TOA summarises the first three ratios: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. The remaining three ratios are their reciprocals.
Trigonometry Class 10 Table of Standard Values
| Ratio | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | Undefined |
| cosec θ | Undefined | 2 | √2 | 2/√3 | 1 |
| sec θ | 1 | 2/√3 | √2 | 2 | Undefined |
| cot θ | Undefined | √3 | 1 | 1/√3 | 0 |
To remember the sine row, use √0/2, √1/2, √2/2, √3/2 and √4/2 for the angles in order. Reverse the sine values to obtain the cosine row. Then calculate tangent as sine divided by cosine wherever that quotient is defined.
Trigonometric Identities Class 10
Three Main Identities
sin2 θ + cos2 θ = 1
1 + tan2 θ = sec2 θ
1 + cot2 θ = cosec2 θ
sin2 θ means (sin θ)2, not sin(θ2). An identity holds for every angle for which all its expressions are defined.
Why Does sin² θ + cos² θ Equal 1?
By Pythagoras’ theorem, O2 + A2 = H2. Dividing by H2 gives (O/H)2 + (A/H)2 = 1. These ratios are sine and cosine, so the identity follows.
Dividing that identity by cos2 θ gives 1 + tan2 θ = sec2 θ. Dividing by sin2 θ gives 1 + cot2 θ = cosec2 θ, provided the respective denominator is non-zero.
Introduction to Trigonometry Class 10 Questions with Solutions
Example 1: Find Other Ratios from sin θ
Question: If sin θ = 5/13 and θ is acute, find cos θ and tan θ.
Take opposite side = 5k and hypotenuse = 13k, where k > 0.
Adjacent side = √[(13k)2 − (5k)2]
Adjacent side = √(144k2)
Adjacent side = 12k
cos θ = 12k/13k
cos θ = 12/13
tan θ = 5k/12k
tan θ = 5/12
Example 2: Evaluate a Standard-Angle Expression
Question: Evaluate 2sin 30° + cos 60° − tan 45°.
2sin 30° + cos 60° − tan 45° = 2 × 1/2 + 1/2 − 1
= 1 + 1/2 − 1
= 1/2
Example 3: Use an Identity to Find a Ratio
Question: If tan θ = 3/4 and θ is acute, find sec θ.
sec2 θ = 1 + tan2 θ
sec2 θ = 1 + 9/16
sec2 θ = 25/16
Since θ is acute, sec θ is positive.
sec θ = 5/4
How to Prove Trigonometric Identities
Begin with one side and transform it into the other using valid identities and algebra. The more complicated side is often a useful starting point, but the best choice depends on the expression. Do not assume the statement you are trying to prove.
- Look for sin² θ + cos² θ or a related rearrangement.
- Convert tan, cot, sec and cosec into sine and cosine when helpful.
- Take a common denominator for fractions.
- Factor expressions before cancelling common factors.
- Check that any cancelled or divided quantity is non-zero.
Example 4: Prove an Identity
Prove: (sec2 θ − 1)/sec2 θ = sin2 θ.
LHS = tan2 θ/sec2 θ
LHS = [sin2 θ/cos2 θ] ÷ [1/cos2 θ]
LHS = sin2 θ
LHS = RHS
Therefore, the identity is proved wherever the original expressions
are defined.
Cancel Factors, Not Terms
You cannot cancel sin θ from the numerator sin θ + cos θ and a denominator containing sin θ. Cancellation applies to common multiplicative factors. Factorise first, or split a fraction correctly when appropriate.
Trigonometry Class 10 Mind Map and Revision 2026
Use the Introduction to Trigonometry Class 10 mind map to connect side ratios, reciprocals, standard values and identities. Cover one branch and reconstruct it from memory. Then explain why a particular formula applies to a worked example.
For 2026 revision, practise identifying sides before learning values and proving identities. When using NCERT solutions for Exercises 8.1, 8.2 or 8.3, match the question with your textbook edition. Older resources may refer to Exercise 8.4 or organise topics differently.
Extra Questions and MCQs with Answers
-
If cos θ = 8/17 and θ is acute, find sin θ.
Answer: 15/17. -
Evaluate sin 60°/cos 30°.
Answer: 1. -
Simplify sec2 θ − tan2 θ.
Answer: 1, where the ratios are defined. -
If cot θ = 7/24 and θ is acute, find cosec θ.
Answer: 25/24.
Introduction to Trigonometry Class 10 MCQ
1. Which ratio equals opposite side ÷ adjacent side?
A. sin θ B. cos θ C. tan θ D. sec θ
Answer: C.
2. What is the value of tan 90°?
A. 0 B. 1 C. 90 D. Undefined
Answer: D. Its denominator cos 90° equals zero.
Related Notes and Worksheet Practice
Practise ratios, values and identities using the Introduction to Trigonometry Class 10 Worksheet with Solutions . Attempt the questions before checking the working.
Revise the geometry behind side ratios with Triangles Class 10 Notes and Mind Map , or explore the CBSE Class 10 Maths Notes and Mind Maps collection .
Frequently Asked Questions
Does this PDF contain Trigonometry Class 10 notes and a mind map?
Yes. The PDF combines notes for understanding the chapter with a mind map for visual revision. Read the explanations first, then recall the ratios, standard values and identities through the map.
How do I identify the opposite and adjacent sides?
Mark the reference angle first. The opposite side faces that angle. The adjacent side touches it but is not the hypotenuse. The hypotenuse is always opposite the right angle.
Why do similar right triangles give the same trigonometric ratios?
Right triangles with the same acute angle are similar by AA. Their corresponding sides scale by the same factor, so the scale factor cancels in each ratio. The ratio depends on the angle rather than the triangle’s size.
How can I remember the Trigonometry Class 10 table?
Learn the sine values using √0/2 through √4/2. Reverse that row for cosine, then obtain tangent by dividing sine by cosine. Use reciprocals for cosec, sec and cot, remembering that division by zero is undefined.
What are the three main trigonometric identities?
They are sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ and 1 + cot² θ = cosec² θ. Each is used only where its expressions are defined.
Why is tan 90° undefined rather than infinity?
tan θ = sin θ/cos θ. At 90°, this becomes 1/0, which is undefined. Infinity is not a real-number value assigned to this division.
Can sin θ or cos θ be greater than 1?
No, for real angles. In an acute right-triangle setting, both ratios are positive and less than 1 because each leg is shorter than the hypotenuse. At the standard endpoints, a ratio may equal 0 or 1.
Why can sec θ be greater than 1?
For an acute angle, sec θ is hypotenuse divided by adjacent side. The numerator is longer than the denominator, so the ratio exceeds 1.
What is the difference between an identity and an equation?
An identity holds for every angle in its domain. An equation may hold only for particular angles. For example, sin² θ + cos² θ = 1 is an identity, while sin θ = 1/2 restricts the possible angles.
What should I do when I get stuck proving an identity?
Compare the form of the two sides. Look for an identity that replaces a squared expression, convert ratios to sine and cosine, or take a common denominator. Write one valid step at a time and avoid unnecessary expansion.
Does sin(A + B) equal sin A + sin B?
No. For example, sin(30° + 30°) = √3/2, but sin 30° + sin 30° = 1. Trigonometric functions do not distribute over addition in this way.
Where can I practise Trigonometry Class 10 questions with solutions?
Use the Introduction to Trigonometry Class 10 worksheet with solutions after revising the notes and mind map. Practise both numerical questions and identity proofs independently.
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