SOLUTION: Sec A + tan A =2 find sin A + cos A ?

Algebra ->  Trigonometry-basics -> SOLUTION: Sec A + tan A =2 find sin A + cos A ?       Log On


   

Question 1109042: Sec A + tan A =2 find sin A + cos A ?

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
sec%28A%29%2Btan%28A%29=2
1%2Fcos%28A%29%2Bsin%28A%29%2Fcos%28A%29=2
%281%2Bsin%28A%29%29%2Fcos%28A%29=2
1%2Bsin%28A%29=2cos%28A%29
sin%28A%29=2cos%28A%29-1
Let us say cos%28A%29=x .
Then,
sin%28A%29=%22+%22+%2B-+sqrt%281-x%5E2%29
Substituting those expressions for sine and cosine,
%22+%22+%2B-+sqrt%281-x%5E2%29=2x-1
Squaring both sides,
1-x%5E2=4x%5E2%2B4x%2B1
0=4x%5E2%2B4x%2B1%2Bx%5E2-1
0=5x%5E2%2B4x
%285x-4%29x=0%0D%0AThe+solutions+are+%7B%7B%7Bx=0 and x=4%2F5=0.8 ,
which translate into cos%28x%29=0 and cos%28x%29=4%2F5=0.8 .
cos%28x%29=0 is not a solution of the original equation,
because it would make sec%28A%29 and tan%28A%29 undefined.
For cos%28A%29=x=4%2F5=0.8 , sec%28A%29=1%2F0.8=1.25 , and
sin%28A%29=%22+%22+%2B-+sqrt%281-x%5E2%29=%22+%22+%2B-+3%2F5=%22+%22+%2B-+0.6 .
sin%28A%29=-0.6 would make tan%28A%29=-0.6%2F0.8=-0.75 and sec%28A%29%2Btan%28A%29=1.25-0.75=0.5 so that is not a solution either.
system%28cos%28A%29=0.8%2Csin%28A%29=0.6%29 makes sec%28A%29%2Btan%28A%29=1.25%2B0.75=2and is the solution.
So sin%28A%29%2Bcos%28A%29=0.8%2B0.6=highlight%281.4%29 .