Introduction to Trigonometry
Mathematics Class Tenth
NCERT Exercise 8.2 Solution part3
Question (3) If tan (A+B) = 3 and tan (A – B) = 1/3; 0o < A + B ≤ 90o; A > B, find A and B.
Solution:
Given, tan ( A + B ) = 3
⇒ tan (A + B) = tan 60o
[∵ tan 60o = 3]
⇒ A + B = 60o - - - - - (i)
Again given, tan (A – B) = 1/√3
⇒ tan(A – B) = tan 30o
[∵ tan 30o = 1/√3]
= A – B = 30o - - - - - (ii)
After addition of equation (i) and (ii) we get
A + B + A – B = 60o + 30o
⇒ 2A = 90o
⇒ A = 90o/2 = 45o
Now after substituting the value of A in equation (i). we get
45o + B = 60o
⇒ B = 60o – 45o
⇒ B = 15o
Thus, A = 45o and B = 15o Answer
Question (4) State whether the following are true or false. Justify your answer.
(i) sin (A + B) = sin A + sin B
Answer: False
Explanation: (a)
sin (A + B) does not mean sin multiplied (×) by (A+B), rather it is the sin of ∠ (A + B)
Thus, sin (A+B) ≠ sin A + sin B
Explanation: (b)
By taking different values of A and B it can be easily proved that sin (A+B) ≠ sin A + sin B
Example:
Let, A = 30o and B = 60o
Thus, sin (A + B)
= sin (30o + 60o)
= sin 90o = 1
And, sin A + sin B
= sin 30o + sin 60o
= 1/2 + 3/2
= 1 + 3/2
Thus, sin (A+B) ≠ sin A + sin B Proved
Thus given statement is False
(ii) The value of sin θ increases as θ increases.
Answer: True
Explanation:
sin 0o = 0
sin 30o = 1/2
sin 45o = 1/2
sin 90o = 1
Clearly, value of sin θ increases as θ increases. And hence given statement is True.
Explanation: 2
We know that sin θ = Side opposite to the angle θ/Hypotenuse = Perpendicular/Hypotenuse
Thus, in the case, when θ = 0, the hypotenuse will coicide with base, i.e. side adjancent to the angle θ and lenght of the perpendicular will become zero.
But with the increase of the angle θ the lenght of perpendicular increase.
Thus with the increase of the lenght of the perpendicular, the value of sin θ will increase.
Clearly, value of sin θ increases as θ increases. And hence given statement is True.
(iii) The value of cos θ increases as θ increases.
Answer: False
Explanation:
cos 0o = 1
cos 30o = 3/2
`os 45o = 1/2
cos 90o = 0
Clearly, value of sin θ decreases as θ increases. And hence given statement is False
Explanation: 2
We know that cos θ = Side adjacent to the angle θ/Hypotenuse = Base/Hypotenuse
Thus, in the case, when θ = 0, the hypotenuse will coicide with base, i.e. side adjancent to the angle θ and lenght of the hypotenuse will become equal to the base.
But with the increase of the angle θ the lenght of hyptenuse increases while the lenght of the base remains same. This means when we observe the cos θ = b/h, the value of h (hypotnuse) which is denominator will increase while the value of numerator, i.e. base will remain same. And, when value of numerator will increase the the value of cose θ will decrease.
Thus with the increase of the lenght of the hypotenuse, the value of cos θ will decrease.
Clearly, value of cos θ decreases as θ increases. And hence given statement is False.
(iv) sin θ = cos θ for all values of θ
Answer: False
Explanation:
sin θ = cos theta; is for only the θ = 45o. Because sin 45o = cos 45o = 2
Thus, given statement is false.
Explanation: 2
The value of sin θ increases with increase the value of θ while the value of cos θ decreases with increase the value of θ
Thus, the given statement "sin θ = cos θ for all values of θ" is false.
(v) cot A is not defined for A = 0o
Answer: True.
Explanation:
We know that,
cot A = cos A/sin A
Thus, let θ = 0
⇒ cot 0o = cos 0o/sin 0o
= 1/0 = ∞ = Undefined
Thus, the given statement "cot A is not defined for A = 0o" is True
Reference: