Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Find the values of other five trigonometric functions of the following:

cotx space equals space 3 over 4 comma space space straight x lies in third quadrant


Answer

Here, cotx space equals space 3 over 4
sinx<0,  cosx<0,  tanx>0, cosecx<0,  secx<0,  cotx>0             (∵ x lies in quadrant III)
Now,                     cosec squared straight x space equals space 1 plus cot squared straight x space space rightwards double arrow space space space space cosec squared straight x space equals space 1 plus 9 over 16 space equals space 25 over 16 space space space rightwards double arrow space space space space cosecx space equals space plus-or-minus 5 over 4
rightwards double arrow                      cosecx space equals space minus 5 over 4                                                 (∵ cosecx < 0)
               sinx space equals space 1 over cosecx space equals space fraction numerator 1 over denominator negative 5 divided by 4 end fraction space equals space minus 4 over 5
               space space space cotx space equals space cosx over sinx
rightwards double arrow                space space space space space space space 3 over 4 equals space fraction numerator cosx over denominator left parenthesis negative 4 divided by 5 right parenthesis end fraction space space space or space cosx space equals space minus 3 over 5
                    space space space space space space secx space equals space 1 over cosx space equals space fraction numerator 1 over denominator negative 3 divided by 5 end fraction space equals space minus 5 over 3
                        tanx space equals space 1 over cotx space equals space fraction numerator 1 over denominator 3 divided by 4 end fraction space equals space 4 over 3

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