The number of all possible values of θ , where 0

Question

The number of all possible values of θ , where 0 < θ < π , for which the system of equations ( y + z ) cos 3 θ = ( x yz ) sin 3 θ x sin 3 θ = y 2 c o s 3 θ ​ + z 2 s i n 3 θ ​ ​ and ( x yz ) sin 3 θ = ( y + 2 z ) cos 3 θ + y sin 3 θ have a solution ( x 0 ​ , y 0 ​ , z 0 ​ ) with y 0 ​ z 0 ​  = 0 , is

TestHub · Accepted Answer

Given equations can be written as x sin 3 θ − y cos 3 θ ​ − z cos 3 θ ​ = 0 x sin 3 θ − y 2 cos 3 θ ​ − z 2 sin 3 θ ​ = 0 and x sin 3 θ − y 2 ​ cos 3 θ − z 1 ​ ( cos 3 θ + sin 3 θ ) = 0 ​ Eqs. (ii) and (iii), implies ⇒ ∴ ⇒ ​ 2 sin 3 θ = cos 3 θ + sin 3 θ sin 3 θ = cos 3 θ tan 3 θ = 1 3 θ = 4 π ​ , 4 5 π ​ , 4 9 π ​ or θ = 12 π ​ , 12 5 π ​ , 12 9 π ​ . ​

Mathematics - Trigonometric Equation Question with Solution | TestHub

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