Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Find the principal solutions (0 ≤ x ≤ 2π ) of the following equation:

cos space straight x space equals space minus fraction numerator 1 over denominator square root of 2 end fraction

Answer

                     space space space space space space cosx space equals space minus fraction numerator 1 over denominator square root of 2 end fraction less than 0 space space space space rightwards double arrow space space space space spacex lies in quadrant II or III
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rightwards double arrow                      cos space straight x space equals space cos space 135 degree or cos space 225 degree
rightwards double arrow                       cos space straight x space equals space space cos open parentheses fraction numerator 3 straight pi over denominator 4 end fraction close parentheses or cos space open parentheses fraction numerator 5 straight pi over denominator 4 end fraction close parentheses
Hence, the principal values of x are fraction numerator 3 straight pi over denominator 4 end fraction and fraction numerator 5 straight pi over denominator 4 end fraction  

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