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Introduction:

Trigonometry

Trigonometry is the study of relationships between the sides and angles of a right angled triangle.

Trigonometric Ratios

Trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.
Let ∆ABC be a triangle right angled at B. Then the trigonometric ratios of the angle A in right ∆ABC are defined as follows:
NCERT Solutions For Class 10 Maths Chapter 8 Introduction to Trigonometry Ex 8.1 1
NCERT Solutions For Class 10 Maths Chapter 8 Introduction to Trigonometry Ex 8.1 2

Note:
The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains same.
NCERT Solutions For Class 10 Maths Chapter 8 Introduction to Trigonometry Ex 8.1 3

Trigonometric Ratios for Complementary Angles

sin (90° – A) = cos A
cos (90° – A) = sin A
tan (90° – A) = cot A
cot (90° – A) = tan A
sec (90° – A) = cosec A
cosec (90° – A) = sec A
Note:
Here (90° – A) is the complementary angle of A.

Trigonometric Identities

An equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angle(s) involved.
(i) sin2θ + cos2θ = 1 [for 0° ≤ θ ≤ 90°]
(ii) sec2θ – tan2θ = 1 [for 0° ≤ θ ≤ 90°]
(iii) cosec2θ – cot2θ = 1 [for 0° < θ ≤ 90°]

Question 1.
In ∆ABC right angled at B, AB = 24 cm, BC = 7 cm. Determine:
(i) sin A, cos A
(ii) sin C, cos C
Solution:
NCERT Solutions for Class 10 Maths Chapter 8 Trigonometry Exercise 8.1 Q1

Question 2.
In given figure, find tan P – cot R.
Solution:
NCERT Solutions for Class 10 Maths Chapter 8 Trigonometry Exercise 8.1 PDF Download Q2

Question 3.
If sin A = \frac { 3 }{ 4 } , calculate cos A and tan A.
Solution:
NCERT Solutions for Class 10 Maths Chapter 8 Trigonometry Exercise 8.1 PDF Q3

Question 4.
Given 15 cot A = 8, find sin A and sec A.
Solution:
NCERT Solutions for Class 10 Maths Chapter 8 Trigonometry Exercise 8.1 Free PDF Download Q4

Question 5.
Given sec θ = \frac { 13 }{ 12 } , calculate all other trigonometric ratios.
Solution:
NCERT Solutions for Class 10 Maths Chapter 8 Trigonometry Exercise 8.1 Q5

Question 6.
If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.
Solution:
Exercise 8.1 Class 10 NCERT Solutions Chapter 8 Trigonometry Q6

Question 7.
If cot θ = \frac { 7 }{ 8 }, evaluate:
(i) \frac { \left( 1+sin\theta \right) \left( 1-sin\theta \right) }{ \left( 1+cos\theta \right) \left( 1-cos\theta \right)}
(ii) cot²θ

Exercise 8.1 Class 10 NCERT Solutions Chapter 8 Trigonometry Free PDF Download Q7

Question 8.
If 3 cot A = 4, check whether \frac { 1-tan^{ 2 }A }{ 1+tan^{ 2 }A } = cos² A – sin² A or not.

Exercise 8.1 Class 10 NCERT Solutions Chapter 8 Trigonometry Free PDF Download Q7

Question 9.
In triangle ABC, right angled at B, if tan A = \frac { 1 }{ \surd 3 }, find the value of:
(i) sin A cos C + cos A sin C
(ii) cos A cos C – sin A sin C
Solution:
Trigonometry Class 10 Chapter 8 Exercise 8.1 NCERT Solutions Free PDF Download Q9
Trigonometry Class 10 Chapter 8 Exercise 8.1 NCERT Solutions PDF Q9.1

Question 10.
In ΔPQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.
Solution:
Trigonometry Class 10 Chapter 8 Exercise 8.1 NCERT Solutions Q10

Question 11.
State whether the following statements are true or false. Justify your answer.
(i) The value of tan A is always less than 1.
(ii) sec A = \frac { 12 }{ 5 } for some value of angle A.
(iii) cos A is the abbreviation used for the cosecant of angle A.
(iv) cot A is the product of cot and A.
(v) sin θ = \frac { 4 }{ 3 } for some angle.
Solution:
Trigonometry Class 10 Chapter 8 Exercise 8.1 NCERT Solutions PDF Download Q11

 

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