SOLUTION: If α and β are acute angles such that cos α = 4/5 and tan β = 15/8, find the following. a). sin (a+B) b). cos (a+B) I am not sure how to start this. If you know how to d

Algebra ->  Equations -> SOLUTION: If α and β are acute angles such that cos α = 4/5 and tan β = 15/8, find the following. a). sin (a+B) b). cos (a+B) I am not sure how to start this. If you know how to d      Log On


   



Question 1158676: If α and β are acute angles such that cos α = 4/5 and tan β = 15/8, find the following.
a). sin (a+B)
b). cos (a+B)
I am not sure how to start this. If you know how to do this please show me the process. I want know how to do this! Thank you.

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

If alpha and beta are acute angles such that cos%28alpha%29=+4%2F5 and tan%28beta%29+=+15%2F8, find the following:

a).sin+%28alpha%2Bbeta%29
sin+%28alpha%2Bbeta%29=sin%28alpha%29+cos%28beta%29+%2B+cos%28alpha%29+sin%28beta%29
given:
cos%28alpha%29=+4%2F5+
find sin%28alpha%29 using identity sin%5E2%28alpha%29%2Bcos%5E2%28alpha%29=1,then
sin%5E2%28alpha%29=1-cos%5E2%28alpha%29..........cos%5E2%28alpha%29=%284%2F5%29%5E2=16%2F25
sin%28alpha%29=sqrt%281-16%2F25%29
sin%28alpha%29=sqrt%289%2F25%29
sin%28alpha%29=3%2F5

use given
tan+%28beta%29+=+15%2F8 to find sin%28beta%29 and cos%28beta%29
since tan%28beta%29=opposite%2Fadjacent+

opposite side=>15
adjacent side=>8
hypotenuse=>sqrt%288%5E2%2B15%5E2%29=17
then
sin%28beta%29=15%2F17
cos%28beta%29=8%2F17


so, we now have

sin%28alpha%29=3%2F5
cos%28alpha%29=+4%2F5+
sin%28beta%29=15%2F17
cos%28beta%29=8%2F17


and we can calculate sin+%28alpha%2Bbeta%29 using identity

sin+%28alpha%2Bbeta%29=sin%28alpha%29+cos%28beta%29+%2B+cos%28alpha%29+sin%28beta%29
sin+%28alpha%2Bbeta%29=%283%2F5%29+%288%2F17%29+%2B+%284%2F5+%29+%2815%2F17%29
sin+%28alpha%2Bbeta%29=24%2F85+%2B+12%2F17
sin+%28alpha%2Bbeta%29=84%2F85


b). cos+%28a%2BB%29 using identity
cos+%28a%2BB%29=cos%28a%29+cos%28B%29+-+sin%28a%29+sin%28B%29
cos+%28a%2BB%29=%284%2F5+%29+%288%2F17%29+-+%283%2F5%29%2815%2F17%29
cos+%28a%2BB%29=32%2F85+-+9%2F17
cos+%28a%2BB%29=-13%2F85