SOLUTION: If sinx+cosx=-1/5, and {{{ 3pi/4 }}}≤x≤pi, find the value of cos2x.

Algebra ->  Trigonometry-basics -> SOLUTION: If sinx+cosx=-1/5, and {{{ 3pi/4 }}}≤x≤pi, find the value of cos2x.      Log On


   

Question 1113956: If sinx+cosx=-1/5, and +3pi%2F4+≤x≤pi, find the value of cos2x.
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
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If  sinx + cosx = -1/5,   then, squaring both sides


sin%5E2%28x%29+%2B+2%2Asin%28x%29%2Acos%28x%29+%2B+cos%5E2%28x%29 = %28-1%2F5%29%5E2,   or,  replacing  sin%5E2%28x%29+%2B+cos%5E2%28x%29 by 1

1 + 2*sin(x)*cos(x) = 1%2F25,  and hense

2*sin(x)*cos(x) = 1%2F25-1 = -24%2F25.



Since  2*sin(x)*cos(x) = sin(2x),  you get

sin(2x) = -24%2F25.



Next, since  3pi%2F4 <= x <= pi, you have  for 2x   3pi%2F2 <= 2x <= 2pi,  i.e.  2x lies in QIV.


Therefore,  cos(2x) = sqrt%281-sin%5E2%282x%29%29 = sqrt%281-%28-24%2F25%29%5E2%29 = sqrt%28%2825%5E2-24%5E2%29%2F25%5E2%29 = sqrt%28%28%2825-24%29%2A%2825%2B24%29%29%2F25%5E2%29 = sqrt%2849%2F25%5E2%29 = 7%2F25.


The sign at sqrt is "+" (plus)  since cosine is positive in QIV.


Answer.  cos(2x) = 7%2F25.