SOLUTION: If sinx+cosx=1.2, then what is the value of {{{ sin^3 (x) + cos^3 (x) }}}?

Algebra ->  Trigonometry-basics -> SOLUTION: If sinx+cosx=1.2, then what is the value of {{{ sin^3 (x) + cos^3 (x) }}}?      Log On


   

Question 1113580: If sinx+cosx=1.2, then what is the value of +sin%5E3+%28x%29+%2B+cos%5E3+%28x%29+?
Answer by ikleyn(52884) About Me  (Show Source):
You can put this solution on YOUR website!
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You are given

sin(x) + cos(x) = 1.2.         (1)


Square both sides.  You will get


sin%5E2%28x%29+%2B+2%2Asin%28x%29%2Acos%28x%29+%2B+cos%5E2%28x%29 = 1.2%5E2,   or, replacing  sin%5E2%28x%29+%2B+cos%5E2%28x%29 by 1


1 + 2*sin(x)*cos(x) = 1.44,    which implies


sin(x)*cos(x) = %281.44+-+1%29%2F2 = 0.22.     (2)


Now we are ready to calculate  sin%5E3%28x%29+%2B+cos%5E3%28x%29.


Use the formula  a%5E3+%2B+b%5E3 = %28a%2Bb%29%2A%28a%5E2+-+ab+%2B+b%5E2%29.  It gives you 


sin%5E3%28x%29+%2B+cos%5E3%28x%29 = %28sin%28x%29+%2B+cos%28x%29%29.%28sin%5E2%28x%29+-+sin%28x%29%2Acos%28x%29+%2B+cos%5E2%28x%29%29.


Now replace in this formula  sin(x) + cos(x) by 1.2 (since it is given); next replace  sin%5E2%28x%29+%2B+cos%5E2%28x%29  by 1  and  sin(x)*cos(x)  by 0.22, 

based on (2).  Then you get  sin%5E3%28x%29+%2B+cos%5E3%28x%29  = 1.2*(1-0.22) = 0.936.


Answer.  sin%5E3%28x%29+%2B+cos%5E3%28x%29 = 0.936.