06Hexamples

= More Examples =
 * Solving Trig Equations **

** Example 1 **
... ... ** (a) ** .. Find the general solution of the equation: math . \\ . \qquad \sqrt{8} \cos \big( 2x \big) + 2 = 0 math

....... ** (b) ** .. Use the general solution to find all the solutions in the domain [0, 2p ]


 * Solution:**

... ... ** (a) ** .. ** Find the general solution **

math \\.\\ . \qquad \sqrt{8} \cos \big( 2x \big) + 2 = 0 \\. \\ . \qquad \sqrt{8} \cos \big( 2x \big) =-2 math . math \\ . \qquad \cos \big( 2x \big) = -\dfrac{2}{\sqrt{8}} = -\dfrac{2}{2\sqrt{2}} \\. \\ . \qquad \cos \big( 2x \big) = -\dfrac{1}{\sqrt{2}} math
 * ** First rearrange to make cos(2x) the subject: **


 * ** Now state the general solution (notice the 2x ) using the rule provided: **

math . \qquad 2x=2n\pi \pm \cos^{-1} \left( -\dfrac{1}{\sqrt{2}} \right),\qquad n \in Z math . math . \qquad \textit{Using exact values } \cos^{-1} \left( -\dfrac{1}{\sqrt{2}} \right)=\dfrac{3\pi}{4},\,\dfrac{5\pi}{4},\, \text{etc} \;\{ \textit{but we only need the first one} \} math . math \\ . \qquad 2x=2n\pi \pm \dfrac{3\pi}{4} \quad \{ \textit{divide entire equation by 2} \} \\. \\ . \qquad x=n\pi \pm \dfrac{3\pi}{8} \qquad n \in Z math

... ... ** (b) ** .. ** Now state the solutions in the domain [0, 2 p ] **


 * ** Substitute various values of n into the general solution. **

math \\ . \qquad n= 0 \quad \Rightarrow \quad 0 \pm \dfrac{3\pi}{8} \quad \Rightarrow \quad -\dfrac{3\pi}{8}, \;\; \dfrac{3\pi}{8} \\. \\ . \qquad n= 1 \quad \Rightarrow \quad \pi \pm \dfrac{3\pi}{8} \quad \Rightarrow \quad \dfrac{5\pi}{8}, \;\; \dfrac{11\pi}{8} \\. \\ . \qquad n= 2 \quad \Rightarrow \quad 2\pi \pm \dfrac{3\pi}{8} \quad \Rightarrow \quad \dfrac{13\pi}{8}, \;\; \dfrac{19\pi}{8} math


 * ** Eliminate the end values because they are outside the required domain **

... ... Solutions are: math . \\ . \qquad x = \left\{ \dfrac{3\pi}{8}, \;\; \dfrac{5\pi}{8}, \;\; \dfrac{11\pi}{8} \;\; \dfrac{13\pi}{8} \right\} math .