find the sum

Question

what is the sum of sin50 cos30 + cos50 sin30

Answer 1

The expression looks like one the addition and subtract angle identities

 

sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

 

x = 50

y = 30

 

 

We can write the expression as

 

sin(50 + 30)

 

Add the angles, we obtain

 

sin(80).

 

 

We are not done yet.  We write this expression again, but as a difference in angles.

 

sin(90 - 10) = sin(90)cos(10) - cos(90)sin(10)

 

 

We know what sin(90) and cos(90) equal to.

 

sin(90) = 1

cos(90) = 0

 

 

Therefore,

 

sin(50) cos(30) + cos(50) sin(30) = sin(90 - 10)

                                                 

                                                  = cos(10)

Answer 2

sin(A+B)=sinAcosB+cosAsinB

working backwards...

sin50cos30+cos50sin30=sin(50+30)

sin(50+30)=sin(80)

sin(80)=0.98480775301 (round as needed)