Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Find the principal and general solutions of the following equation:

cot space straight x space equals space minus square root of 3




Answer
cot space space straight x space equals space minus square root of 3 space space space space space rightwards double arrow space space tan space straight x space equals space minus fraction numerator 1 over denominator square root of 3 end fraction is -ve
rightwards double arrow x lies in quadrant II or IV.
∴                              tan space straight x space equals space minus fraction numerator 1 over denominator square root of 3 end fraction space equals space minus space tan space 30 degree space equals space tan left parenthesis 180 degree minus 30 degree right parenthesis space space or space tan space left parenthesis 360 degree space minus space 30 degree right parenthesis

                                               equals space tan space 150 degree space or space tan space 330 degree
rightwards double arrow                             tan space straight x space equals space tan open parentheses fraction numerator 5 straight pi over denominator 6 end fraction close parentheses space space or space space tan open parentheses fraction numerator 11 straight pi over denominator 6 end fraction close parentheses

∴   The principal solutions are:  fraction numerator 5 straight pi over denominator 6 end fraction comma space space fraction numerator 11 straight pi over denominator 6 end fraction
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