Find sin (A+B)

Question

If:

sin A = 3/5

and

tan B = 4/3

Answer 1

sin(A + B) = sinAcosB + cosAsinB

 

sin A is given as 3/5 and tan B is given as 4/3.  That tells us that we have a 3-4-5 right triangle.  Draw it (I tried but could not get it to come out looking right on the computer display).

 

Now you can see that

 

sin A = 3/5

 

sin B = 4/5

 

cos A = 4/5

 

cos B = 3/5

 

sin(A + B) = sinAcosB + cosAsinB  =  3/5 times 3/5  plus  4/5 times 4/5  = 9/25  +   16/25  =  25/25  = 1

 

Alternate (aft.drawing your triangle):

Note that angle A is the complement of angle B: measure of angle A + measure of angle B = 90 degrees.

Sin (A + B) = Sin (90) = 1

 

 

Answer 2

We can use the sum identity.

 

sinA cosB  -  cosA sinB

 

 

Next, draw a right triangle and label it using the given information.  We will have a 3,4,5 triangle.  From the triangle, we can find cosB, cos A, and sinB.

 

cosB = 3/5

 

cosA = 4/5

 

sin B = 4/5

 

 

Substitute these values into the identity.

 

 
sinA cosB  -  cosA sinB =

 

(3/5)(3/5) - (4/5)(4/5) =

 

(9/25) - (16/25) =

 

-7/25 =

 

-7/25