Calculate tg 2a if a is a real number so sin a + cos a = 1

Expert Answers
hala718 eNotes educator| Certified Educator

sina + cosa =1

Square both side:

(sin a + cos a )^2 = 1

sin^2 a + 2sinacosa + cos^2 a = 1

We know that sin^2 a + cos^2 a=1

==> 1+ 2sina*cosa =1

==> 2sina*cosa=0

We know that 2sina*cosa= sin2a

==> 2sin2a=0

==> sin2a= 0

==> 2a= n*pi  n=0,1,2 ...

==> cos2a= cos(npi) = +-1

tg 2a= +-npi

 

giorgiana1976 | Student

sin a + cos a=1

If we'll divide both sides by cos a, we'll get:

sin a/cos a + 1= 1/cos a

tg a + 1= 1/cos a

tg a= (1/cos a) -1

tg 2a = tg (a+a) = (tg a+ tga)/[1-(tga)^2]

tg 2a= 2tga/[1-(tga)^2]

tg 2a=2(1/cosa -1)/{1-[(1/cosa)+1]{1+ [(1/cosa)-1]}

tg 2a= 2(1-cos a)/(cosa )*[(2 - 1/cosa)(1/cos a)]

tg 2a = 2(cos a)(1-cosa)/(2cosa - 1)

neela | Student

To calculate tan2a such that sina+cosa = 1.

solution:

sina+cosa = 1.

So sina +sqrt(1-sina)^2 = 1. Or

 sqrt(1-sin(a)^2 ) = 1-sina. Squaring both sides, we get:

1-sin^2 = 1+2sia+(sina)^2. Or

2sina +2sin^2 = 0. Or

sina(1+sina) =  0.. Or

sina = 0. Or a = npi. Or

sina = -1 Or a = -pi/2 + npi.

Therefore tan2a = tan(2npi) = 0. Or

tan2a = tan 2(-pi/2+ npi) = tan (2npi -pi) = 0.