SOLUTION: If {{{ sin x + cos x = -( 1 / 5 ) }}}, and 3pi/4 < or = x < or = pi, find the value of cos2x. Should be solved without the use of a calculator.

Algebra ->  Trigonometry-basics -> SOLUTION: If {{{ sin x + cos x = -( 1 / 5 ) }}}, and 3pi/4 < or = x < or = pi, find the value of cos2x. Should be solved without the use of a calculator.       Log On


   

Question 1177381: If +sin+x+%2B+cos+x+=+-%28+1+%2F+5+%29+, and 3pi/4 < or = x < or = pi, find the value of cos2x. Should be solved without the use of a calculator.
Answer by ikleyn(52802) About Me  (Show Source):
You can put this solution on YOUR website!
.
If sin(x) + cos(x) = - 1%2F5, and 3pi%2F4 <= x <= pi, find the value of cos2x.
Should be solved without the use of a calculator.
~~~~~~~~~~~~~~~~ 


Square both sides


    sin%5E2%28x%29 + 2%2Asin%28x%29%2Acos%28x%29 + cos%5E2%28x%29 = %28-1%2F5%29%5E2


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

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

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


Hence,  cos(2x) = sqrt%281+-+sin%5E2%282x%29%29 = sqrt%281-%28-24%2F25%29%5E2%29 = sqrt%281-576%2F625%29 = sqrt%28%28625-576%29%2F625%29 = sqrt%2849%2F625%29 = +/- 7%2F25.


Since  3pi%2F4 <= x <= pi,  we have  3pi%2F2 <= 2x <= 2pi.


In other words, the angle 2x is in QIV.


Cosine is positive in QIV,  therefore our answer is  cos(2x) = + 7%2F25.

Solved, and the answer is obtained without using the calculator, as requested.