CBSE Class 10 Mathematics • Chapter 8
Introduction to Trigonometry Class 10 Worksheet with Solutions
Practise trigonometric ratios, standard angle values and identities. Follow worked examples, attempt extra questions and use the formula sheet and FAQs to strengthen your understanding.
This Introduction to Trigonometry Class 10 worksheet helps you understand the relationship between the sides and angles of a right triangle. Begin by identifying the opposite side, adjacent side and hypotenuse. Once these are clear, selecting the correct ratio becomes much easier.
Use the worksheet available on this page alongside the original solved examples and practice questions below. These Introduction to Trigonometry Class 10 extra questions with answers cover ratio calculations, standard values and identity proofs. Attempt each question independently before checking its solution.
Introduction to Trigonometry: Basic Concepts
For an acute angle θ in a right triangle, the hypotenuse is the side opposite the right angle. The opposite side lies across from θ, while the adjacent side is the other side touching θ.
The Reference Angle Matters
In △ABC, right-angled at B, AC is the hypotenuse. For angle A, BC is the opposite side and AB is the adjacent side. For angle C, AB becomes the opposite side and BC becomes the adjacent side.
The hypotenuse stays the same, but the opposite and adjacent sides change when you change the reference angle.
Introduction to Trigonometry Class 10 Formula Sheet
| Ratio | Definition for an Acute Angle θ |
|---|---|
| sin θ | Opposite / Hypotenuse |
| cos θ | Adjacent / Hypotenuse |
| tan θ | Opposite / Adjacent |
| cosec θ | Hypotenuse / Opposite |
| sec θ | Hypotenuse / Adjacent |
| cot θ | Adjacent / Opposite |
Reciprocal and Quotient Relationships
cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
tan θ = sin θ/cos θ
cot θ = cos θ/sin θ
Use these relationships wherever the denominators are non-zero. For acute angles, all six ratios are defined and positive.
Introduction to Trigonometry Class 10 Table of Standard Values
Learn the values for 0°, 30°, 45°, 60° and 90°. The endpoint values at 0° and 90° extend the acute-angle ratios; they do not describe an ordinary non-degenerate right triangle with those angles as its acute angles.
| 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 | Not defined |
| cosec θ | Not defined | 2 | √2 | 2/√3 | 1 |
| sec θ | 1 | 2/√3 | √2 | 2 | Not defined |
| cot θ | Not defined | √3 | 1 | 1/√3 | 0 |
Introduction to Trigonometry Class 10 Identities
A trigonometric identity is true for every angle for which its expressions are defined. The three main identities are:
sin2 θ + cos2 θ = 1
1 + tan2 θ = sec2 θ
1 + cot2 θ = cosec2 θ
Here, sin2 θ means (sin θ)2. It does not mean sin(θ2).
Useful rearrangements include 1 − sin2 θ = cos2 θ, sec2 θ − tan2 θ = 1 and cosec2 θ − cot2 θ = 1.
Class 10 Trigonometry Examples with Solutions
Example 1: Find All Six Ratios
Question: In △ABC, right-angled at B, AB = 12 cm and BC = 5 cm. Find the trigonometric ratios of angle A.
By Pythagoras’ theorem:
AC2 = AB2 + BC2
AC2 = 144 + 25
AC2 = 169
AC = 13 cm
For angle A, opposite = 5, adjacent = 12 and hypotenuse = 13.
sin A = 5/13
cos A = 12/13
tan A = 5/12
cosec A = 13/5
sec A = 13/12
cot A = 12/5
Example 2: Find Ratios from tan θ
Question: If tan θ = 3/4 and θ is acute, find sin θ and cos θ.
Let opposite = 3k and adjacent = 4k, where k > 0.
Hypotenuse = √[(3k)2 + (4k)2]
Hypotenuse = 5k
sin θ = 3k/5k
sin θ = 3/5
cos θ = 4k/5k
cos θ = 4/5
Example 3: Evaluate Using Standard Values
Question: Evaluate 2sin 30° cos 60° + tan2 45°.
= 2 × (1/2) × (1/2) + 12
= 1/2 + 1
= 3/2
Example 4: Prove an Identity
Question: Prove that (1 − cos2 θ)/sin θ = sin θ, where sin θ ≠ 0.
Start with the left-hand side:
LHS = (1 − cos2 θ)/sin θ
LHS = sin2 θ/sin θ
LHS = sin θ
LHS = RHS
The identity is proved using 1 − cos2 θ = sin2 θ.
Example 5: Simplify a Fraction Using Identities
Question: Prove tan2 θ/(sec θ + 1) = sec θ − 1 for an acute angle θ.
LHS = (sec2 θ − 1)/(sec θ + 1)
LHS = [(sec θ − 1)(sec θ + 1)]/(sec θ + 1)
LHS = sec θ − 1
LHS = RHS
First use tan2 θ = sec2 θ − 1. Then apply the algebraic identity a2 − b2 = (a − b)(a + b).
Introduction to Trigonometry Class 10 MCQ with Answers
1. What is sin 30°?
A. 1 B. 1/2 C. √3/2 D. 0
Answer: B. 1/2.
2. Which expression equals 1?
A. sin θ + cos θ
B. sin2 θ + cos2 θ
C. tan2 θ − sec2 θ
Answer: B. This is the fundamental trigonometric identity.
3. If θ is acute and tan θ = 1, what is θ?
A. 30° B. 45° C. 60° D. 90°
Answer: B. 45°.
4. What is sec2 θ − tan2 θ?
A. 0 B. 1 C. −1 D. 2
Answer: B. 1. Rearrange 1 + tan2 θ = sec2 θ.
Introduction to Trigonometry Class 10 Extra Questions with Answers
- If sin θ = 8/17 and θ is acute, find cos θ and tan θ.
- Evaluate sin2 60° + cos2 30°.
- If sec θ = 13/12 and θ is acute, find tan θ.
- Simplify (1 − sin2 θ)/cos2 θ, where cos θ ≠ 0.
- Prove that 1/(tan θ + cot θ) = sin θ cos θ for an acute angle θ.
Check the Practice Answers
- cos θ = 15/17 and tan θ = 8/15.
- 3/4 + 3/4 = 3/2.
- tan θ = 5/12.
- The expression equals 1.
- tan θ + cot θ = (sin2 θ + cos2 θ)/(sin θ cos θ) = 1/(sin θ cos θ). Taking the reciprocal gives the required result.
How to Practise NCERT Trigonometry Exercises
For Class 10 Maths Introduction to Trigonometry Exercise 8.1, focus on identifying sides and calculating ratios. For Exercise 8.2, practise standard angle values. For identity questions, including Exercise 8.3 in editions that place identities there, learn to transform one side step by step.
Exercise organisation can differ between editions. If your textbook includes Introduction to Trigonometry Exercise 8.4, match solutions to the exact question. Use NCERT solutions, NCERT Exemplar problems and verified previous year questions alongside this worksheet for further practice.
This chapter provides the foundation for Class 10 Applications of Trigonometry, where ratios are used to calculate heights and distances. Strengthen ratio selection and standard values before attempting those applications.
Common Trigonometry Mistakes to Avoid
- Choosing opposite and adjacent sides without fixing the angle.
- Confusing sin θ with its reciprocal, cosec θ.
- Writing sin2 θ as sin(θ2).
- Assuming sin θ + cos θ always equals 1.
- Cancelling terms across addition instead of cancelling factors.
- Using a reciprocal when its denominator is zero.
- Substituting a single angle value when asked to prove an identity.
FAQs on Introduction to Trigonometry Class 10
What is trigonometry in Class 10?
Trigonometry studies relationships between angles and sides. Class 10 introduces six ratios in a right triangle, standard angle values and identities used to simplify expressions.
How do I remember sin, cos and tan?
Use SOH–CAH–TOA: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse and tangent = opposite/adjacent. Identify the reference angle before labelling the sides.
Why do trigonometric ratios depend on the angle rather than triangle size?
Right triangles with the same acute angle are similar by AA. Their corresponding sides are proportional, so the ratios remain unchanged when the triangle is enlarged or reduced.
Can sin θ or cos θ be greater than 1 for an acute angle?
No. In a right triangle, the hypotenuse is longer than either other side. Therefore, the opposite/hypotenuse and adjacent/hypotenuse ratios lie between 0 and 1. However, tan θ can exceed 1; for example, tan 60° = √3.
Why is tan 90° not defined?
tan θ = sin θ/cos θ. At 90°, sin 90° = 1 and cos 90° = 0. Division by zero is undefined, so tan 90° is not defined.
How do I prove a trigonometric identity?
Start with one side, usually the more complicated expression. Use reciprocal relationships, the three main identities and algebraic factorisation to transform it into the other side. Show each equality clearly and keep denominators non-zero.
Does checking an identity at 30° prove it?
No. Checking one angle confirms only that particular case. An identity proof must establish the equality for every angle in its domain using valid algebraic steps.
What are the three main Introduction to Trigonometry Class 10 identities?
They are sin2 θ + cos2 θ = 1, 1 + tan2 θ = sec2 θ and 1 + cot2 θ = cosec2 θ. The expressions must be defined at the angle used.
How can I find all ratios when one ratio is given?
For an acute angle, represent the known ratio using proportional side lengths. Find the missing side with Pythagoras’ theorem, then calculate the remaining ratios. For tan θ = 3/4, use sides 3k, 4k and 5k.
Is sin(A + B) equal to sin A + sin B?
No, this is not a general identity. For example, sin(30° + 30°) = √3/2, while sin 30° + sin 30° = 1. Trigonometric functions do not distribute over addition.
How should I use this worksheet with NCERT solutions?
Attempt each question first. Check your side selection, ratio, standard values and identity transformations against the solution. Reattempt incorrect questions without looking at the answer.
Where can I find an Introduction to Trigonometry Class 10 mind map?
Visit the Introduction to Trigonometry Notes and Mind Map on WitKnowLearn. Use it to review the chapter before returning to this worksheet for practice.
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