Q. 42 C

Find the value of

sin A. cos B + cos A. sin B. if and tan B = √3.

Answer :

Given:





Let,


Side opposite to angle A =BC = 1k


Side adjacent to angle A =AB = k√3


where, k is any positive integer


Firstly we have to find the value of BC.


So, we can find the value of AC with the help of Pythagoras theorem


(AB)2 + (BC)2 = (AC)2


(√3k)2 + (1k)2 = (AC)2


(AC)2 = 1 k2 +3 k2


(AC)2 = 4 k2


AC =2 k2


AC =±2k


But side AC can’t be negative. So, AC = 2k


Now, we will find the sin A and cos A



Side opposite to angle A = BC = k


and Hypotenuse = AC = 2k


So,


Now, we know that,



Side adjacent to angle A = AB =k√3


Hypotenuse = AC =2k


So,


Now,



Given: tan B = √3




Let,


Side opposite to angle B =AC = √3k


Side adjacent to angle B =AB = 1k


where, k is any positive integer


Firstly we have to find the value of BC.


So, we can find the value of AC with the help of Pythagoras theorem


(AB)2 + (AC)2 = (BC)2


(1k)2 + (√3k)2 = (BC)2


(BC)2 = 1 k2 +3 k2


(BC)2 = 4 k2


BC =2 k2


BC =±2k


But side BC can’t be negative. So, BC = 2k


Now, we will find the sin B and cos B



Side opposite to angle B = AC = k√3


and Hypotenuse = BC = 2k


So,


Now, we know that,



Side adjacent to angle B = AB =1k


Hypotenuse = BC =2k


So,


Now, sin A. cos B + cos A. sin B


Putting the values of sin A, sin B cos A and Cos B, we get





=1


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