Question

if A;B are acute angles tanA=5/12;cosB=3/5thencos(A+B)=​

Answer 1

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Answer 2

Therefore the value of Cos ( A + B ) = 16/65.

Given:

Tan A = 5 / 12 and Cos B = 3 / 5

Where A and B are the acute angles.

To Find:

The value of Cos ( A + B )

Solution:

The given problem can be solved very easily as shown below.

We know that ( 5, 12, 13 ) and ( 3, 4, 5 ) are Pythagoras triplets.

So hypotenuse should be 13 and 5 respectively in both triangles.

For angle A:

Tan A = Opposite side / Adjacent side = 5 / 12

So Opposite side = 5

Adjacent side = 12

Hypotenuse = 13

⇒ Sin A = Opposite side / Hypotenuse = 5 / 13

⇒ Cos A = Adjacent side / Hypotenuse = 12 / 13

For angle B:

Cos B = ( Adjacent side / Hypotenuse )  = 3 / 5

Adjacent side = 3

Opposite side = 4

Hypotenuse = 5

⇒ Sin B = ( Opposite side ) / ( Hypotenuse ) = 4 / 5

Now, Cos ( A + B ) = Cos A. Cos B - Sin A. Sin B

Where Cos A = 12/13; Cos B = 3/5; Sin A = 5/13; Sin B = 4/5

⇒ Cos ( A + B ) = ( 12/13 ) × ( 3/5 ) - ( 5/13 ) × ( 4/5 )

⇒ Cos ( A + B ) = 36/65 - 20/65 = 16/65

Therefore the value of Cos ( A + B ) = 16/65.

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