tutorial exercise use inverse functions where needed to find all solutions of the equation in the interval…

tutorial exercise use inverse functions where needed to find all solutions of the equation in the interval 0, 2π). tan²x - 8 tan x - 9 = 0 step 1 begin by treating the given equation as a quadratic in tan x and factor. tan²x - 8 tan x - 9 = 0 (tan x + 1)(tan x - 9) = 0 step 2 set each factor equal to zero. use the inverse tangent function to obtain the solutions in the interval 0, 2π). (enter your answers as a comma - separated list.) tan x + 1 = 0 or tan x - 9 = 0 tan x = -1 or tan x = 9 x = tan⁻¹(3π/4), tan⁻¹(7π/4) × submit skip (you cannot come back) submit answer

tutorial exercise use inverse functions where needed to find all solutions of the equation in the interval 0, 2π). tan²x - 8 tan x - 9 = 0 step 1 begin by treating the given equation as a quadratic in tan x and factor. tan²x - 8 tan x - 9 = 0 (tan x + 1)(tan x - 9) = 0 step 2 set each factor equal to zero. use the inverse tangent function to obtain the solutions in the interval 0, 2π). (enter your answers as a comma - separated list.) tan x + 1 = 0 or tan x - 9 = 0 tan x = -1 or tan x = 9 x = tan⁻¹(3π/4), tan⁻¹(7π/4) × submit skip (you cannot come back) submit answer

Answer

Explanation:

Step1: Factor the quadratic - like equation

Let $t = \tan x$. The equation $\tan^{2}x - 8\tan x-9 = 0$ becomes $t^{2}-8t - 9=0$. Factoring $t^{2}-8t - 9$, we get $(t + 1)(t - 9)=0$. Substituting back $t=\tan x$, we have $(\tan x + 1)(\tan x - 9)=0$.

Step2: Solve for $\tan x$

Set each factor equal to zero:

  • For $\tan x+1 = 0$, we have $\tan x=-1$. In the interval $[0,2\pi)$, the solutions of $\tan x=-1$ are $x=\frac{3\pi}{4},\frac{7\pi}{4}$ since $\tan x$ has a period of $\pi$ and $\tan^{-1}(-1)=-\frac{\pi}{4}$, and adding $\pi$ to get solutions in the desired interval.
  • For $\tan x - 9=0$, we have $\tan x = 9$. Then $x=\tan^{-1}(9)+k\pi,k\in\mathbb{Z}$. In the interval $[0,2\pi)$, the solutions are $x=\tan^{-1}(9),\tan^{-1}(9)+\pi$.

Answer:

$\frac{3\pi}{4},\frac{7\pi}{4},\tan^{-1}(9),\tan^{-1}(9)+\pi$