Find the value of

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

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