NCERT Solutions For Class 10 Maths Chapter 8 Introduction to Trigonometry Ex 8.1
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:
Ex 8.1 Class 10 Maths Question 2.
In given figure, find tan P – cot R.
Solution:
Ex 8.1 Class 10 Maths Question 3.
If sin A =
Solution:
Ex 8.1 Class 10 Maths Question 4.
Given 15 cot A = 8, find sin A and sec A.
Solution:
Ex 8.1 Class 10 Maths Question 5.
Given sec θ =
Solution:
Ex 8.1 Class 10 Maths Question 6.
If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.
Solution:
Ex 8.1 Class 10 Maths Question 7.
If cot θ =
(i)
(ii) cot²θ
Solution:
Ex 8.1 Class 10 Maths Question 8.
If 3 cot A = 4, check whether
Solution:
Ex 8.1 Class 10 Maths Question 9.
In triangle ABC, right angled at B, if tan A =
(i) sin A cos C + cos A sin C
(ii) cos A cos C – sin A sin C
Solution:
Ex 8.1 Class 10 Maths 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:
Ex 8.1 Class 10 Maths 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 =
(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 =
Solution:
Class 10 Maths Introduction To Trigonometry
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:
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.
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°]
NCERT Solutions For Class 10 Maths Chapter 8 Introduction to Trigonometry Ex 8.2
Ex 8.2 Class 10 Maths Question 1.
Evaluate the following:
Solution:
Ex 8.2 Class 10 Maths Question 2.
Choose the correct option and justify your choice:
Solution:
Ex 8.2 Class 10 Maths Question 3.
If tan (A + B) = √3 and tan (A – B) =
Solution:
Ex 8.2 Class 10 Maths Question 4.
State whether the following statements are true or false. Justify your answer.
(i) sin (A + B) = sin A + sin B.
(ii) The value of sin θ increases as θ increases.
(iii) The value of cos θ increases as θ increases.
(iv) sin θ = cos θ for all values of θ.
(v) cot A is not defined for A = 0°.
Solution:
NCERT Solutions For Class 10 Maths Chapter 8 Introduction to Trigonometry Ex 8.3
Ex 8.3 Class 10 Maths Question 1.
Evaluate:
Solution:
Ex 8.3 Class 10 Maths Question 2.
Show that:
(i) tan 48° tan 23° tan 42° tan 67° = 1
(ii) cos 38° cos 52° – sin 38° sin 52° = 0
Solution:
Ex 8.3 Class 10 Maths Question 3.
If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A.
Solution:
Ex 8.3 Class 10 Maths Question 4.
If tan A = cot B, prove that A + B = 90°.
Solution:
Ex 8.3 Class 10 Maths Question 5.
If sec 4A = cosec (A – 20°), where 4A is an acute angle, find the value of A.
Solution:
Ex 8.3 Class 10 Maths Question 6.
If A, B and C are interior angles of a triangle ABC, then show that: sin (
Solution:
Ex 8.3 Class 10 Maths Question 7.
Express sin 61° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.
Solution:
NCERT Solutions For Class 10 Maths Chapter 8 Introduction to Trigonometry Ex 8.4
Ex 8.4 Class 10 Maths Question 1.
Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.Solution:
Ex 8.4 Class 10 Maths Question 2.
Write all the other trigonometric ratios of ∠A in terms of sec A.
Solution:
Ex 8.4 Class 10 Maths Question 3.
Evaluate:
Solution:
Ex 8.4 Class 10 Maths Question 4.
Choose the correct option. Justify your choice.
(i) 9 sec² A – 9 tan² A = ……
(A) 1
(B) 9
(C) 8
(D) 0
(ii) (1 + tan θ + sec θ) (1 + cot θ – cosec θ) = ………..
(A) 0
(B) 1
(C) 2
(D) -1
(iii) (sec A + tan A) (1 – sin A) = ………….
(A) sec A
(B) sin A
(C) cosec A
(D) cos A
(iv)
(A) sec² A
(B) -1
(C) cot² A
(D) tan² A
Solution:
Ex 8.4 Class 10 Maths Question 5.
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
Solution:
