Question 7.6.3: Solving a Trigonometric Equation (Zero-Factor Property) Solv...

\tan² x + \tan x – 2 = 0                          This equation is quadratic in form.

(\tan x – 1)(\tan x + 2) = 0                     Factor.

\tan x – 1 = 0    \text{or}      \tan x + 2 = 0       Zero-factor property

\tan x = 1          \text{or}    \tan x = -2              Solve each equation.

The solutions for \tan x = 1 over the interval [0, 2π) are x = \frac{π}{4} and x = \frac{5π}{4} .

To solve \tan x = -2 over that interval, we use a calculator set in radian mode. We find that \tan^{-1}(-2) ≈ -1.1071487. This is a quadrant IV number, based on the range of the inverse tangent function. However, because we want solutions over the interval [0, 2π), we must first add π to -1.1071487, and then add 2π.

See Figure 32.

x ≈ -1.1071487 + π ≈ 2.0344439

x ≈ -1.1071487 + 2π ≈ 5.1760366

The solutions over the required interval form the following solution set.

\left\{ \underbrace{ \frac{\pi }{4},               \frac{5}{4} , }_{\begin{matrix} \text{Exact }\\ \text{values }\end{matrix} } \underbrace{2.0344,          5.1760 }_{\begin{matrix} \text{Approximate values to} \\ \text{four decimal places} \end{matrix} } \right\}