Question Number 102190 by dw last updated on 07/Jul/20

Find the value of  x ∙∙[Trigonometric Subs.]  x−(x/(√(1−x^2 )))=((91)/(60))

Answered by bemath last updated on 07/Jul/20

((x((√(1−x^2 ))−1))/(√(1−x^2 ))) = ((91)/(60))  let x = sin θ  ((sin θ.(cos θ−1))/(cos θ)) = ((91)/(60))  60sin θcos θ−60sin θ=91cos θ

Answered by 1549442205 last updated on 07/Jul/20

Put x=cosϕ(0≤ϕ≤π)we have   cosϕ−((cosϕ)/(sinϕ))=((91)/(60))⇔60cosϕsinϕ−60cosϕ=91sinϕ(1)  Put t=tan(ϕ/2).We get  (1)⇔60((2(1−t^2 )t)/((1+t^2 )^2 ))−60((1−t^2 )/(1+t^2 ))=91((2t)/(1+t^2 ))  ⇔120(t−t^3 )−60(1−t^4 )=182(t^3 +t)  ⇔30t^4 −151t^3 −31t−30=0  ⇔30t^4 −151t^3 −31t−30=0   we get two roots(by Calculator)  t_1 =−0.45781901488,t_2 =5.08098312713  From this we get  ϕ_1 =2arctan t_1 =−49°11′54 (is rejected  as ϕ_1 outside [0;π])  ϕ_2 =2arctan t_2 =157°43′53  Hence,x_2 =cosϕ_2 =−0.9254187445  Thus,the unique root x=−0.925417445  satisfying our prpblem