SOLUTION: Suppose that {{{a^2 + b^2=1}}} and {{{c^2 + d^2=1}}}. Prove: {{{abs(ac + bd)<= 1}}}. Hint: Let {{{a= cos(theta)}}}, {{{b=sin(theta)}}}, {{{c= cos(phi)}}}, and {{{d=sin(phi)}}}.

Algebra ->  Trigonometry-basics -> SOLUTION: Suppose that {{{a^2 + b^2=1}}} and {{{c^2 + d^2=1}}}. Prove: {{{abs(ac + bd)<= 1}}}. Hint: Let {{{a= cos(theta)}}}, {{{b=sin(theta)}}}, {{{c= cos(phi)}}}, and {{{d=sin(phi)}}}.      Log On


   

Question 968326: Suppose that a%5E2+%2B+b%5E2=1 and c%5E2+%2B+d%5E2=1. Prove: abs%28ac+%2B+bd%29%3C=+1.
Hint: Let a=+cos%28theta%29, b=sin%28theta%29, c=+cos%28phi%29, and d=sin%28phi%29.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Suppose that a%5E2+%2B+b%5E2=1 and c%5E2+%2B+d%5E2=1. Prove: abs%28ac+%2B+bd%29%3C=+1.
Hint: Let a=+cos%28theta%29, b=sin%28theta%29, c=+cos%28phi%29, and d=sin%28phi%29. 

abs%28ac+%2B+bd%29%3C=+1

abs%28cos%28theta%29cos%28phi%29+%2B+sin%28theta%29sin%28phi%29%29%3C=+1

abs%28cos%28theta-phi%29%29%3C=+1

All cosines are between -1 and +1.  So the above is true.

Edwin