Processing Math: Done
Solution 4.4:2d
From Förberedande kurs i matematik 1
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- | {{ | + | Apart from the fact that there is a |
- | < | + | <math>\text{5}x</math>, this is a normal trigonometric equation of the type |
- | {{ | + | <math>\text{sin }y\text{ }=a</math> . If we are only interested in solutions which satisfy |
- | { | + | <math>0\le \text{5}x\le \text{2}\pi </math>, then a sketch of the unit circle shows that there are two such solutions, |
- | + | <math>\text{5}x=\text{ }\frac{\pi }{4}</math> | |
- | {{ | + | and the reflectionally symmetric solution |
+ | <math>\text{5}x=\text{ }\pi \text{-}\frac{\pi }{4}=\frac{3\pi }{4}</math>. | ||
+ | |||
[[Image:4_4_2_d.gif|center]] | [[Image:4_4_2_d.gif|center]] | ||
+ | |||
+ | All of the equation's solutions are obtained from all values of 5x which differ by a multiple of 2π from either of these two solutions: | ||
+ | |||
+ | |||
+ | <math>\text{5}x=\text{ }\frac{\pi }{4}+2n\pi </math> | ||
+ | and | ||
+ | <math>\text{5}x=\frac{3\pi }{4}+2n\pi </math>, | ||
+ | |||
+ | where | ||
+ | <math>n</math> | ||
+ | is an arbitrary integer. | ||
+ | |||
+ | If we divide both of these by | ||
+ | <math>\text{5}</math>, we obtain the solutions expressed in terms of x alone: | ||
+ | |||
+ | |||
+ | <math>x=\frac{\pi }{20}+\frac{2}{5}n\pi </math> | ||
+ | and | ||
+ | <math>x=\frac{3\pi }{20}+\frac{2}{5}n\pi </math>, | ||
+ | |||
+ | where | ||
+ | <math>n</math> | ||
+ | is an arbitrary integer. |
Revision as of 13:55, 30 September 2008
Apart from the fact that there is a
5x
2
4
-
4=43
All of the equation's solutions are obtained from all values of 5x which differ by a multiple of 2π from either of these two solutions:
4+2n
+2n
where
If we divide both of these by
20+52n
+52n
where