Question #155165

Solve for θ to the nearest hundredth, where 0 ≤ x ≤ 2π. a) 3 tan2 θ – 2 tan θ = 1


1
Expert's answer
2021-01-19T04:38:41-0500

Solution


  • First write the tan2θ\tan2\theta in terms of tanθ\tan\theta to get the equation in like terms.
  • Then,

3(2tanθ1tan2θ)2tanθ=1\qquad\qquad \begin{aligned} 3\bigg(\frac{2\tan\theta}{1-\tan^2\theta}\bigg)-2\tan\theta&=1\\ \end{aligned}

  • For easy recognition, substitute tanθ=x\tan\theta=x & simplify the equation to find the roots of it.
  • Then,

6x1x22x=02x3+x2+4x11x2=0\qquad\qquad \begin{aligned} \frac{6x}{1-x^2}-2x&=0\\ \frac{2x^3+x^2+4x-1}{1-x^2}&=0 \end{aligned}

  • Now x±1x \neq \pm1 for the equation to exist.
  • Then its about finding the roots of the numerator equation.
  • They are

x={0.23060.365+1.427i0.3651.426ix=\begin{cases} 0.2306\\ -0.365+1.427i\\ -0.365-1.426i \end{cases}


  • All those values are acceptable only for an algebraic variable like x but only 0.2306 is valid for tanθ\tan \theta as <tanθ<+-\infty<\tan\theta<+\infty
  • Therefore,

x=tanθ=0.2306θ=0.2266rad\qquad\qquad \begin{aligned} x&=\tan\theta=0.2306\\ \theta&= 0.2266 rad \end{aligned}



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS