Inverse Trigonometric Functions

Write in the simplest form. - Wired Faculty

Question

Write $tan to the power of negative 1 end exponent open square brackets fraction numerator 3 straight a squared space straight x minus straight x cubed over denominator straight a left parenthesis straight a cubed minus 3 straight x squared right parenthesis end fraction close square brackets comma space straight a greater than 0 comma space space minus fraction numerator straight a over denominator square root of 3 end fraction less or equal than straight x less or equal than fraction numerator straight a over denominator square root of 3 end fraction$ in the simplest form.

Solution

Let space space straight theta equals tan to the power of negative 1 end exponent open square brackets fraction numerator 3 straight a squared space straight x minus straight x cubed over denominator straight a left parenthesis straight a squared minus 3 straight x squared right parenthesis end fraction close square brackets equals tan open square brackets fraction numerator 3 begin display style straight x over straight a end style minus begin display style straight x cubed over straight a cubed end style over denominator 1 minus begin display style fraction numerator 3 space straight x squared over denominator straight a squared end fraction end style end fraction close square brackets

( Dividing num. and deno by a³)
Put

straight x over straight a equals tan space a

therefore space space space space space space space space space space space space straight theta equals tan to the power of negative 1 end exponent open square brackets fraction numerator 3 space tan space straight a space minus space tan cubed space straight a over denominator 1 minus 3 space tan squared space straight a end fraction close square brackets equals tan to the power of negative 1 end exponent space left parenthesis tan space 3 straight a right parenthesis space equals space 3 space straight a therefore space space space space space space space space space space space space straight theta space equals space 3 space tan to the power of negative 1 end exponent space straight x over straight a