Processing Math: 94%
To print higher-resolution math symbols, click the
Hi-Res Fonts for Printing button on the jsMath control panel.
No jsMath TeX fonts found -- using image fonts instead.
These may be slow and might not print well.
Use the jsMath control panel to get additional information.
jsMath Control PanelHide this Message

jsMath

Solution 4.4:8c

From Förberedande kurs i matematik 1

(Difference between revisions)
Jump to: navigation, search
m (Lösning 4.4:8c moved to Solution 4.4:8c: Robot: moved page)
Line 1: Line 1:
-
{{NAVCONTENT_START}}
+
When we have a trigonometric equation which contains a mixture of different trigonometric functions, a useful strategy can be to rewrite the equation so that it is expressed in terms of just one of the functions. Sometimes, it is not easy to find a way to rewrite it, but in the present case a plausible way is to replace the “
-
<center> [[Image:4_4_8c-1(2).gif]] </center>
+
<math>\text{1}</math>
-
{{NAVCONTENT_STOP}}
+
” in the numerator of the left-hand side with
-
{{NAVCONTENT_START}}
+
<math>\text{sin}^{\text{2}}x+\text{cos}^{\text{2}}x\text{ }</math>
-
<center> [[Image:4_4_8c-2(2).gif]] </center>
+
using the Pythagorean identity. This means that the equation's left-hand side can be written as
-
{{NAVCONTENT_STOP}}
+
 
 +
 
 +
<math>\frac{1}{\cos ^{2}x}=\frac{\cos ^{2}x+\sin ^{2}x}{\cos ^{2}x}=1+\frac{\sin ^{2}x}{\cos ^{2}x}=1+\tan ^{2}x</math>
 +
 
 +
 
 +
and the expression is then completely expressed in terms of tan x,
 +
 
 +
 
 +
<math>1+\tan ^{2}x=1-\tan x</math>
 +
 
 +
 
 +
If we substitute
 +
<math>t=\tan x</math>
 +
, we see that we have a second-degree equation in
 +
<math>t</math>
 +
, which, after simplifying, becomes
 +
<math>t^{\text{2}}\text{ }+t=0</math>
 +
and has roots
 +
<math>t=0</math>
 +
and
 +
<math>t=-\text{1}</math>. There are therefore two possible values for
 +
<math>\tan x</math>,
 +
<math>\tan x=0</math>
 +
tan x =0 or
 +
<math>\tan x=-1</math>
 +
The first equality is satisfied when
 +
<math>x=n\pi </math>
 +
for all integers
 +
<math>n</math>, and the second when
 +
<math>x=\frac{3\pi }{4}+n\pi </math>.
 +
 +
The complete solution of the equation is
 +
 
 +
 +
<math>\left\{ \begin{array}{*{35}l}
 +
x=n\pi \\
 +
x=\frac{3\pi }{4}+n\pi \\
 +
\end{array} \right.</math>
 +
(
 +
<math>n</math>
 +
an arbitrary integer).

Revision as of 08:14, 2 October 2008

When we have a trigonometric equation which contains a mixture of different trigonometric functions, a useful strategy can be to rewrite the equation so that it is expressed in terms of just one of the functions. Sometimes, it is not easy to find a way to rewrite it, but in the present case a plausible way is to replace the “ 1 ” in the numerator of the left-hand side with sin2x+cos2x using the Pythagorean identity. This means that the equation's left-hand side can be written as


1cos2x=cos2xcos2x+sin2x=1+sin2xcos2x=1+tan2x


and the expression is then completely expressed in terms of tan x,


1+tan2x=1−tanx


If we substitute t=tanx , we see that we have a second-degree equation in t , which, after simplifying, becomes t2 +t=0 and has roots t=0 and t=−1. There are therefore two possible values for tanx, tanx=0 tan x =0 or tanx=−1 The first equality is satisfied when x=n for all integers n, and the second when x=43+n.

The complete solution of the equation is


x=nx=43+n  ( \displaystyle n an arbitrary integer).

Retrieved from "http://wiki.math.se/wikis/2008/bridgecourse1-ImperialCollege/index.php/Solution_4.4:8c"