Q. 42 A

Find the value of

cos A sin B + sin A. cos B, if sin A= 4/5 and cos B = 12/13.

Answer :



Given:


To find: cos A sin B + sin A cos B


As, we have the value of sin A and cos B but we don’t have the value of cos A and sin B


So, First we find the value of cos A and sin B



We know that,



Or



Let,


Side opposite to angle A = 4k


and Hypotenuse = 5k


where, k is any positive integer


So, by Pythagoras theorem, we can find the third side of a triangle


(P)2 + (B)2 = (H)2


(4k)2 + (B)2 = (5)2


16 k2 + (B)2 = 25 k2


(B)2 = 25 k2 –16 k2


(B)2 = 9 k2


B =9 k2


B =±3k [taking positive square root since, side cannot be negative]


So, Base = 3k


Now, we have to find the value of cos A


We know that,



Side adjacent to angle A =3k


Hypotenuse =5k


So,


Now, we have to find the sin B



We know that,




Let,


Side adjacent to angle B =12k


Hypotenuse =13k


where, k is any positive integer


So, by Pythagoras theorem, we can find the third side of a triangle


(B)2 + (P)2 = (H)2


(12k)2 + (P)2 = (13)2


144 k2 + (P)2 = 169 k2


(P)2 = 169 k2 –144 k2


(P)2 = 25 k2


P =25 k2


P =±5k [taking positive square root since, side cannot be negative]


So, Perpendicular = 5k


Now, we have to find the value of sin B


We know that,




Now, cos A sin B + sin A cos B


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





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