Solution 4.4:6c

From Förberedande kurs i matematik 1

(Difference between revisions)
Jump to: navigation, search
m (Lösning 4.4:6c moved to Solution 4.4:6c: Robot: moved page)
Line 1: Line 1:
-
{{NAVCONTENT_START}}
+
If we use the trigonometric relation
-
<center> [[Image:4_4_6c.gif]] </center>
+
<math>\text{sin }\left( -x \right)=-\text{sin }x</math>, the equation can be rewritten as
-
{{NAVCONTENT_STOP}}
+
 
 +
 
 +
<math>\sin 2x=\sin \left( -x \right)</math>
 +
 
 +
 
 +
In exercise 4.4:5a, we saw that an equality of the type
 +
 
 +
 
 +
<math>\sin u=\sin v\quad </math>
 +
 
 +
 
 +
is satisfied if
 +
 
 +
 
 +
<math>u=v+2n\pi </math>
 +
or
 +
<math>u=\pi -v+2n\pi </math>
 +
 
 +
 
 +
where
 +
<math>n\text{ }</math>
 +
is an arbitrary integer. The consequence of this is that the solutions to the equation satisfy
 +
 
 +
 
 +
<math>2x=-x+2n\pi </math>
 +
or
 +
<math>2x=\pi -\left( -x \right)+2n\pi </math>
 +
 
 +
 
 +
i.e.
 +
 
 +
 
 +
<math>3x=2n\pi </math>
 +
or
 +
<math>x=\pi +2n\pi </math>
 +
 
 +
 
 +
The solutions to the equation are thus
 +
 
 +
 +
<math>\left\{ \begin{array}{*{35}l}
 +
x=\frac{2n\pi }{3} \\
 +
x=\pi +2n\pi \\
 +
\end{array} \right.</math>
 +
(
 +
<math>n\text{ }</math>
 +
an arbitrary integer)

Revision as of 11:54, 1 October 2008

If we use the trigonometric relation \displaystyle \text{sin }\left( -x \right)=-\text{sin }x, the equation can be rewritten as


\displaystyle \sin 2x=\sin \left( -x \right)


In exercise 4.4:5a, we saw that an equality of the type


\displaystyle \sin u=\sin v\quad


is satisfied if


\displaystyle u=v+2n\pi or \displaystyle u=\pi -v+2n\pi


where \displaystyle n\text{ } is an arbitrary integer. The consequence of this is that the solutions to the equation satisfy


\displaystyle 2x=-x+2n\pi or \displaystyle 2x=\pi -\left( -x \right)+2n\pi


i.e.


\displaystyle 3x=2n\pi or \displaystyle x=\pi +2n\pi


The solutions to the equation are thus


\displaystyle \left\{ \begin{array}{*{35}l} x=\frac{2n\pi }{3} \\ x=\pi +2n\pi \\ \end{array} \right. ( \displaystyle n\text{ } an arbitrary integer)

Retrieved from "http://wiki.math.se/wikis/2008/bridgecourse1-ImperialCollege/index.php/Solution_4.4:6c"