Processing Math: Done

From Förberedande kurs i matematik 1

Revision as of 12:44, 3 April 2008 (edit) Oskar (Talk | contribs) (Ny sida: {{NAVCONTENT_START}} <center> Bild:4_4_7c-1(3).gif </center> {{NAVCONTENT_STOP}} {{NAVCONTENT_START}} <center> Bild:4_4_7c-2(3).gif </center> {{NAVCONTENT_STOP}} {{NAVCONTENT_START}...) ← Previous diff		Current revision (07:59, 14 October 2008) (edit) (undo) Tek (Talk | contribs) m
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-	{{NAVCONTENT_START}}	+	If we want to solve the equation <math>\cos 3x = \sin 4x</math>, we need an additional result which tells us for which values of ''u'' and ''v'' the equality
-	<center> [[Bild:4_4_7c-1(3).gif]] </center>	+	<math>\cos u = \sin v</math> holds, but to get that we have to start with the equality <math>\cos u=\cos v</math>.
-	{{NAVCONTENT_STOP}}	+
-	{{NAVCONTENT_START}}	+	So, we start by looking at the equality
-	<center> [[Bild:4_4_7c-2(3).gif]] </center>	+
-	{{NAVCONTENT_STOP}}	+	{{Displayed math||<math>\cos u=\cos v\,\textrm{.}</math>}}
-	{{NAVCONTENT_START}}	+
-	<center> [[Bild:4_4_7c-3(3).gif]] </center>	+	We know that for fixed ''u'' there are two angles <math>v=u</math> and <math>v=-u</math> in the unit circle which have the cosine value <math>\cos u</math>, i.e. their ''x''-coordinate is equal to <math>\cos u\,</math>.
-	{{NAVCONTENT_STOP}}	+
		+	[[Image:4_4_7_c1.gif|center]]
		+
		+	Imagine now that the whole unit circle is rotated anti-clockwise an angle <math>\pi/2</math>. The line <math>x=\cos u</math> will become the line <math>y=\cos u</math> and the angles ''u'' and -''u'' are rotated to <math>u+\pi/2</math> and <math>-u+\pi/2</math>, respectively.
		+
		+	[[Image:4_4_7_c2.gif|center]]
		+
		+	The angles <math>u+\pi/2</math> and <math>-u+\pi/2</math> therefore have their ''y''-coordinate, and hence sine value, equal to <math>\cos u</math>. In other words, the equality
		+
		+	{{Displayed math||<math>\cos u = \sin v</math>}}
		+
		+	holds for fixed ''u'' in the unit circle when <math>v = \pm u + \pi/2</math>, and more generally when
		+
		+	{{Displayed math||<math>v = \pm u + \frac{\pi}{2} + 2n\pi\qquad</math>(''n'' is an arbitrary integer).}}
		+
		+	For our equation <math>\cos 3x = \sin 4x</math>, this result means that ''x'' must satisfy
		+
		+	{{Displayed math||<math>4x = \pm 3x + \frac{\pi}{2} + 2n\pi\,\textrm{.}</math>}}
		+
		+	This means that the solutions to the equation are
		+
		+	{{Displayed math||<math>\left\{\begin{align}
		+	x &= \frac{\pi}{2} + 2n\pi\,,\\[5pt]
		+	x &= \frac{\pi}{14} + \frac{2}{7}\pi n\,,
		+	\end{align}\right.</math>}}
		+
		+	where ''n'' is an arbitrary integer.

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Current revision

If we want to solve the equation cos3x=sin4x, we need an additional result which tells us for which values of u and v the equality cosu=sinv holds, but to get that we have to start with the equality cosu=cosv.

So, we start by looking at the equality

cosu=cosv.

We know that for fixed u there are two angles v=u and v=−u in the unit circle which have the cosine value cosu, i.e. their x-coordinate is equal to cosu.

Imagine now that the whole unit circle is rotated anti-clockwise an angle 2. The line x=cosu will become the line y=cosu and the angles u and -u are rotated to u+2 and −u+2, respectively.

The angles u+2 and −u+2 therefore have their y-coordinate, and hence sine value, equal to cosu. In other words, the equality

cosu=sinv

holds for fixed u in the unit circle when v=u+2, and more generally when

v=u+2+2n(n is an arbitrary integer).

For our equation cos3x=sin4x, this result means that x must satisfy

4x=3x+2+2n.

This means that the solutions to the equation are

xx=2+2n=14+72n

where n is an arbitrary integer.