Question 7.6.4: Solving a Trigonometric Equation (Quadratic Formula) Find al...
Solving a Trigonometric Equation (Quadratic Formula)
Find all solutions of \cot x(\cot x + 3) = 1.
We multiply the factors on the left and subtract 1 to write the equation in standard quadratic form.
\cot x(\cot x + 3) = 1 Original equation
\cot² x + 3 \cot x – 1 = 0 Distributive property; Subtract 1.
This equation is quadratic in form, but cannot be solved using the zero-factor property. Therefore, we use the quadratic formula, with a = 1, b = 3, c = -1, and \cot x as the variable.
\cot x =\frac{ -b ± \sqrt{b² – 4ac }}{2a} Quadratic formula
= \frac{-3 ± \sqrt{3² – 4(1)(-1)}}{ 2(1)} a = 1, b = 3, c = -1
↑
\fbox{Be careful with signs.}= \frac{-3 ±\sqrt{9 + 4 }}{2} Simplify.
= \frac{-3 ± \sqrt{13}}{ 2} Add under the radical.
\cot x ≈ -3.302775638 \text{or} \cot x ≈ 0.3027756377Use a calculator.
x ≈ \cot^{-1}(-3.302775638) \text{or} x ≈ \cot^{-1}(0.3027756377)Definition of inverse cotangent
x ≈ \tan^{-1}(\frac{1}{ -3.302775638 }) +π \text{or} x ≈ \tan^{-1}( \frac{1}{0.3027756377} )Write inverse cotangent in terms of
inverse tangent.
x ≈ -0.2940013018 + π or x ≈ 1.276795025
Use a calculator in radian mode.
x ≈ 2.847591352
To find all solutions, we add integer multiples of the period of the tangent function, which is π , to each solution found previously. Although not unique, a common form of the solution set of the equation, written using the least possible nonnegative angle measures, is given as follows.
{2.8476 + nπ , 1.2768 + nπ , where n is any integer}
Round to four decimal places.