If 𝛼 and 𝛽 are acute angles such that cos 𝛼 = 15/17 and tan 𝛽 = 3/4 , find the following.

Question

(A) sin(𝛼 + 𝛽)

(B)  cos(𝛼 + 𝛽)

Answer 1

sin( a + b ) = sin( a ) cos( b ) + cos( a ) sin( b )

cos( a + b ) = cos( a ) cos( b ) - sin( a ) sin( b )

'a' and 'b' are acute angles meaning they stay in Quadrant 1, meaning you are able to take all squareroots to be the positive solution.

sin( a ) = √( 1 - ( cos( a ) )² )

cos( b ) = 1 / ( √( 1 - ( tan( b ) )² ) )

These angles fall into 2 special right triangles:

8-15-17 triangle & 3-4-5 triangle

Meaning:

sin( a ) = 8/17 (you can double check thru the formula above)

cos( b ) = 4/5 & sin( b ) = 3/5

Plugging into the sum trig equations yields:

sin( a + b ) = ( 8/17 )( 4/5 ) + ( 15/17 )( 3/5 ) = ( 32 + 45 )/85 sin( a + b ) = 77/85

cos( a + b ) = ( 15/17 )( 4/5 ) - ( 8/17 )( 3/5 ) = ( 60 - 24 )/85 cos( a + b ) = 36/85

I hope this helps! Message me in the comments if you have any questions or if I did anything wrong!