Principle of Mathematical Induction

Principle of Mathematical Induction

Question

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#6 {main}</pre> in the following:

cosx space equals space minus 1 third comma space straight x is in quadrant III

Answer
cosx space equals space minus 1 third  (Given) x lies in quadrant III
rightwards double arrow                   straight pi less than straight x less than fraction numerator 3 straight pi over denominator 2 end fraction   rightwards double arrow       straight pi over 2 less than straight x over 2 less than fraction numerator 3 straight pi over denominator 4 end fraction

∴  straight x over 2 lies in quadrant II.
therefore space space cos straight x over 2 space and space tan straight x over 2 are -ve, but sin straight x over 2 is +ve
Now,             cos open parentheses straight x over 2 close parentheses space equals space minus square root of fraction numerator 1 plus cosx over denominator 2 end fraction end root space equals space minus square root of fraction numerator 1 minus begin display style 1 third end style over denominator 2 end fraction end root space rightwards double arrow space space   cos open parentheses straight x over 2 close parentheses space equals space minus square root of 1 third end root
                    sin open parentheses straight x over 2 close parentheses space equals space plus square root of fraction numerator 1 minus cosx over denominator 2 end fraction end root space equals space square root of fraction numerator 1 plus begin display style 1 third end style over denominator 2 end fraction end root    rightwards double arrow   sin open parentheses straight x over 2 close parentheses space equals space square root of 2 over 3 end root
                   tan open parentheses straight x over 2 close parentheses space equals space fraction numerator sin open parentheses begin display style straight x over 2 end style close parentheses over denominator cos open parentheses begin display style straight x over 2 end style close parentheses end fraction space equals space fraction numerator square root of begin display style 2 over 3 end style end root over denominator negative square root of begin display style 1 third end style end root end fraction             rightwards double arrow    tan open parentheses straight x over 2 close parentheses space equals space minus square root of 2

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