NCERT Solutions for Class 10 Maths Chapter 8 Exercise 8.1 are provided here to help students build a strong foundation in Trigonometry. These solutions are prepared by subject experts according to the latest CBSE syllabus and are designed to explain concepts in a clear, step-by-step manner.
Exercise 8.1 introduces students to the basic trigonometric ratios —
NCERT Solutions Class 10 Maths Chapter 8 Exercise 8.1 – Introduction to Trigonometry
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If A = 60° and B = 30°, verify that : sin(A + B) = sin A cos B + cos A sin B
[CBSE - 2024]
NCERT Solutions Class 10 Maths Chapter 8 Exercise 8.1 – Introduction to Trigonometry
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sin θ (Sine)
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cos θ (Cosine)
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tan θ (Tangent)
Students learn how to define these ratios using a right-angled triangle and understand the relationship between the sides:
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Perpendicular
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Base
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Hypotenuse
By practicing this exercise, students develop a clear understanding of basic trigonometric definitions, which forms the base for solving advanced trigonometry problems in later exercises.
These solutions are aligned with the CBSE exam pattern, helping students prepare effectively and score well in board examinations.
Q1. In ΔABC, right-angled at B, AB = 24 cm, BC = 7 cm, determine
(i) sin A, cos A
(ii) sin C, cos C
Solution:
AC² = AB² + BC²
= 24² + 7²
= 576 + 49
= 625
AC = √625 = 25 cm
(i) For angle A:
sin A = BC / AC = 7 / 25
cos A = AB / AC = 24 / 25
(ii) For angle C:
sin C = AB / AC = 24 / 25
cos C = BC / AC = 7 / 25
Q2. In the given figure, find tan P – cot R.
Given: PQ = 12 cm, PR = 13 cm
Solution:
QR² = PR² − PQ²
= 13² − 12²
= 169 − 144
= 25
QR = 5
tan P = QR / PQ = 5 / 12
cot R = QR / PQ = 5 / 12
tan P − cot R = 0
Q3. If sin A = 3/4, find cos A and tan A.
Solution:
Let BC = 3k and AC = 4k
AB² = AC² − BC²
= 16k² − 9k²
= 7k²
AB = √7 k
cos A = AB / AC = √7 / 4
tan A = BC / AB = 3 / √7
Q4. Given 15 cot A = 8, find sin A and sec A.
Solution:
cot A = 8 / 15
Let AB = 8k, BC = 15k
AC² = 8² + 15²
= 64 + 225
= 289
AC = 17k
sin A = BC / AC = 15 / 17
sec A = AC / AB = 17 / 8
Q5. If sec θ = 13/12, find all other trigonometric ratios.
Solution:
Let AC = 13k, AB = 12k
BC² = AC² − AB²
= 169k² − 144k²
= 25k²
BC = 5k
sin θ = 5 / 13
cos θ = 12 / 13
tan θ = 5 / 12
cot θ = 12 / 5
cosec θ = 13 / 5
Q6. If cos A = cos B and A, B are acute angles, show that A = B.
Solution:
cos A = adjacent / hypotenuse
cos B = adjacent / hypotenuse
Given cos A = cos B
Therefore adjacent sides are equal.
Hence corresponding angles are equal.
∠A = ∠B
Q7. If cot θ = 7/8, evaluate
(i) (1 − sin θ)/(1 + sin θ)
(ii) cot² θ
Solution:
Let AB = 7k, BC = 8k
AC² = 49k² + 64k²
= 113k²
AC = √113 k
sin θ = 8 / √113
cos θ = 7 / √113
(i) (1 − sin θ)/(1 + sin θ) = 49/64
(ii) cot² θ = (7/8)² = 49/64
Q8. If 3 cot A = 4, check whether
(1 − tan²A)/(1 + tan²A) = cos²A − sin²A
Solution:
cot A = 4/3
Let AB = 4k, BC = 3k
AC = 5k
tan A = 3/4
sin A = 3/5
cos A = 4/5
LHS = (1 − 9/16) / (1 + 9/16)
= 7/25
RHS = 16/25 − 9/25
= 7/25
Both sides are equal.
Q9. In ΔABC right-angled at B, tan A = 1/3, find
(i) sin A cos C + cos A sin C
(ii) cos A cos C − sin A sin C
Solution:
Let BC = k, AB = 3k
AC² = 10k²
AC = √10 k
sin A = 1/√10
cos A = 3/√10
sin C = 3/√10
cos C = 1/√10
(i) = 1
(ii) = 0
Q10. In ΔPQR right-angled at Q, PR + QR = 25 cm, PQ = 5 cm.
Solution:
Let PR = x
QR = 25 − x
Using Pythagoras:
x² = 5² + (25 − x)²
x² = 25 + 625 − 50x + x²
50x = 650
x = 13
PR = 13 cm
QR = 12 cm
sin P = 12/13
cos P = 5/13
tan P = 12/5
Q11. State whether true or false
(i) tan A always less than 1 → False
(ii) sec A = 12/5 possible → True
(iii) cos A is abbreviation of cosecant → False
(iv) cot A is product of cot and A → False
(v) sin θ = 4/3 possible → False
FAQs: Class 10 Maths Chapter 8 – Exercise 8.1
Q1. What is the main focus of Exercise 8.1?
Answer:
Exercise 8.1 focuses on understanding and defining the basic trigonometric ratios (sin θ, cos θ, tan θ) in a right-angled triangle.
Q2. What are the basic trigonometric ratios?
Answer:
In a right-angled triangle:
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sin θ = Perpendicular / Hypotenuse
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cos θ = Base / Hypotenuse
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tan θ = Perpendicular / Base
Q3. Why is Exercise 8.1 important?
Answer:
This exercise builds the foundation of trigonometry, which is essential for solving problems in later chapters and higher classes.
Q4. What concepts should I understand before solving Exercise 8.1?
Answer:
You should understand:
Q5. How do NCERT Solutions help in exam preparation?
Answer:
NCERT Solutions provide clear explanations, correct formulas, and step-by-step solutions, helping students understand concepts properly and perform confidently in exams.
If you want, I can also provide:
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Step-by-step solved examples
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Important formulas for quick revision
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Board exam important questions from Chapter 8
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Practice questions with answers
These NCERT Solutions for Class 10 Maths Chapter 8 Exercise 8.1 will help you strengthen your basics in Trigonometry and prepare effectively for your Class 10 board exams.