NCERT Solutions for Class 10 Maths Exercise 8.2 of Chapter 8 Introduction to Trigonometry
NCERT Solutions for Class 10 Maths Exercise 8.2 of Chapter 8 Introduction to Trigonometry are mandatory for the preparation of CBSE board exams of 10 class .You can clear all your doubts about Introduction to Trigonometry by the study of these NCERT Solutions of exercise 8.1 Chapter 8.NCERT Solutions for Class 10 Maths Exercise 8.2 of Chapter 8 Introduction to Trigonometry are created by an expert of maths who has huge experience of maths teaching from class 6 to 12 classes.We have been teaching academic classes from class 1 to 12 in our private coaching centre for last 25 years in New Delhi,therefore you are free to ask us your problems in study we will definitely solve your all obstacles on your ways through our huge experience in the field of education.
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NCERT Solutions for Class 10 Mathsย Chapter 8 Introduction to Trigonometry
Exercise 8.1- Introduction to Trigonometry
Exercise 8.2 -Introduction to Trigonometry
Exercise 8.3-Introduction to Trigonometry
Exercise 8.4-Introduction to Trigonometry
NCERT solutions of Important Questions-Introduction to Trigonometry
NCERT Solutions for Class 10 Maths Exercise 8.2 of Chapter 8 Introduction to Trigonometry
Q1.Evaluate the following
(i) sin 60ยฐcos 30ยฐ+ sin 30ยฐcos 60ยฐ
(ii) 2 tanยฒ45ยฐ+ cosยฒ30ยฐ- sinยฒ 60ยฐ
Ans. (i) sin 60ยฐcos 30ยฐ+ sin 30ยฐcos 60ยฐ
Putting the followingย values in the given expression
sin 60ยฐ=โ3/2,cos 30ยฐ=โ3/2, sin 30ยฐ=1/2,cos 60ยฐ= 1/2
(โ3/2) ร(โ3/2) + (1/2)ร(1/2)
=(3/4) + (1/4)
=4/4 = 1
(ii) 2 tanยฒ45ยฐ+ cosยฒ30ยฐ- sinยฒ 60ยฐ
Putting the values of tan45ยฐ=1, cos30ยฐ=โ3/2, sin 60ยฐ=โ3/2
2ร1ยฒ +(โ3/2)ยฒ -(โ3/2)ยฒ
=2 + 3/4 -3/4 = 2
(iii) cos 45ยฐ/(sec30ยฐ+ cosec 30ยฐ)
Putting the values of cos 45ยฐ = 1/โ2, sec 30ยฐ =2/โ3, cosec 30ยฐ =2 in the given expression
(iv) The given expression is
(sin 30ยฐ + tan 45ยฐ-cosec 60ยฐ)/(sec 30ยฐ + cos 60ยฐ+cot 45ยฐ)
Putting the values of sin 30ยฐ=1/2,ย tan 45ยฐ=1,cosec 60ยฐ=2/โ3,sec 30ยฐ=2/โ3 , cos 60ยฐ=1/2,cot 45ยฐ=1 in the given expression
Rationalizing the denominator
Putting the values of cos 60ยฐ = 1/2, sec 30ยฐ = 2/โ3 , tan 45ยฐ =1, sec 30ยฐ = 2/โ3, cos 30ยฐ = โ3/2
Q2. Choose the correct options and justify your choice.
(i) 2 tan 30ยฐ/(1+ tanยฒ30ยฐ)=
(A) sin 60ยฐย ย (B) cos 60ยฐย ย (C) tan 60ยฐย ย (D) sin 30ยฐ
(ii) (1 – tanยฒ45ยฐ)/ (1 – tanยฒ45ยฐ)=
(A) tan 90ยฐย ย (B) 1ย ย ย (C) sin 45ยฐย (D) 0
Ans (A) is correct
Substitute tan 30ยฐ in the given equation
tan 30ยฐ = 1/โ3
2 tan 30ยฐ/ 1 + tanยฒ 30ยฐ = 2(1/โ3)/1 + (1/โ3)ยฒ
= (2/โ3)/(1 + 1/3) = (2/โ3)/(4/3)
= 6/4โ3 = โ3/2 = sin 60ยฐ
The obtained solution is equivalent to the trigonometric ratio sin 60ยฐ
(ii) D is correct
Substitute tan 45ยฐ in the given equation
tan 45ยฐ = 1
1 – tanยฒ 45ยฐ/1 + tanยฒ 45ยฐ = (1 – 1ยฒ)/(1 + 1ยฒ)
= 0/2 = 0
The solution of the above equation is 0
(iii) (A) is correct
To find the value of A, substitute the degree given in the options one by one sin 2A = 2 sin A is true when A = 0ยฐ
As sin 2A = sin 0ยฐ = 0
2 sin A = 2 sin 0ยฐ = 2 ร 0 = 0
(iv) (C) is correct
Substitite tan 30ยฐ in the given equation
tan 30ยฐ = 1/โ3
2tan30ยฐ/1 – tanยฒ30ยฐ = 2(1/โ3)/1 – (1/โ3)ยฒ
= (2/โ3)/(1 – 1/3) = (2/โ3)/(2/3) = โ3 = tan 60ยฐ
The value of the given equation is equivalent to tan 60ยฐ
Q3. If tan (A + B) = โ3 and tan (A – B) = 1/โ3, 0ยฐ < A + B โค 90ยฐ; A > B, find A and B
Ans. tan (A + B) = โ3
Since โ3ย = tan 60ยฐ
Now substitute the degree value
โ tan (A +B) = tan 60ยฐ
(A + B) = 60ยฐ … (i)
The above equation is assumed as equation (i)
tan (A – B) = 1/โ3
Since 1/โ3 = tan 30ยฐ
Now substitute the degree value
โ tan (A – B) = tan 30ยฐ
(A – B) = 30ยฐ … (ii)
Now add the equation (i) and (ii), we get
A + B + A – B = 60ยฐ + 30ยฐ
Cancel the terms B
2A = 90ยฐ
A = 45ยฐ
Now, substitute the value of A in equation (i) to find the value of B
45ยฐ + B = 60ยฐ
B = 60ยฐ – 45ยฐ
B = 15ยฐ
Therefore A = 45ยฐ and B = 15ยฐ
Q4. State whether the following are true or false. Justify your answer
(i) sin (A + B) = sin A + sin B
(ii) The value of sin ฮธ increases as ฮธ increases
(iii) The value of cos ฮธ increases as ฮธ increases
(iv) sin ฮธ = cos ฮธ for all values of ฮธ
(v) cot A is not defined for A = 0ยฐ
Ans. False
Justification:
Let us take A = 30ยฐ and B = 60ยฐ, then
Substitute the values in the sin (A + B) formula, we get
sin (A + B) = sin (30ยฐ + 60ยฐ) = sin 90ยฐ = 1 and,
sin A + sin B = sin 30ยฐ + sin 6oยฐ
= 1/2 + โ3/2 = 1 + โ3/2
Since the values obtained are not equal, the solution is false
(ii) True
Justification:
According to the values obtained as per the unit circle, the values of sin are:
sin 0ยฐ = 0
sin 30ยฐ = 1/2
sin 45ยฐ = 1/โ2
sin 60ยฐ = โ3/2
sin 90ยฐ = 1
Thus the value of sin ฮธ increases as ฮธ increases. Hence, the statement is true
(ii) False
Justification:
According to the values obtained as per the unit circle, the values of cos are:
cos 0ยฐ = 1
cos 30ยฐ = โ3/2
cos 45ยฐ = 1/โ2
cos 60ยฐ = 1/2
cos 90ยฐ = 0
Thus the value of cos ฮธ increases as ฮธ increases. Hence, the statement is false
(iv) False
sin ฮธ = cos ฮธ, when a right triangle has 2 angles of (ฯ/4). Therefore, the above statement is false
(v) True
Since cot function is the reciprocal of the tan function, it is also written as:
cot A = cos A/sin A
Now substitute A = 0ยฐ
cot 0ยฐ = cos 0ยฐ/sin 0ยฐ = 1/0 = undefined
Hence, it is true
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