From Förberedande kurs i matematik 1

Revision as of 08:09, 3 April 2008 (edit) Oskar (Talk | contribs) (Ny sida: {{NAVCONTENT_START}} <center> Bild:4_2_8-1(2).gif </center> {{NAVCONTENT_STOP}} {{NAVCONTENT_START}} <center> Bild:4_2_8-2(2).gif </center> {{NAVCONTENT_STOP}}) ← Previous diff		Current revision (12:23, 9 October 2008) (edit) (undo) Tek (Talk | contribs) m
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-	{{NAVCONTENT_START}}	+	We start by drawing three auxiliary triangles, and calling the three vertical sides ''x'', ''y'' and ''z'', as shown in the figure.
-	<center> [[Bild:4_2_8-1(2).gif]] </center>	+
-	{{NAVCONTENT_STOP}}	+	[[Image:4_2_8.gif|center]]
-	{{NAVCONTENT_START}}	+
-	<center> [[Bild:4_2_8-2(2).gif]] </center>	+	Using the definition of cosine, we can work out ''x'' and ''y'' from
-	{{NAVCONTENT_STOP}}	+
		+	{{Displayed math||<math>\begin{align}
		+	x &= a\cos \alpha\,,\\[3pt]
		+	y &= b\cos \beta\,,
		+	\end{align}</math>}}
		+
		+	and, for the same reason, we know that ''z'' satisfies the relation
		+
		+	{{Displayed math||<math>z=\ell\cos \gamma\,\textrm{.}</math>}}
		+
		+	In addition, we know that the lengths ''x'', ''y'' and ''z'' satisfy the equality
		+
		+	{{Displayed math||<math>z=x-y\,\textrm{.}</math>}}
		+
		+	If we substitute in the expressions for ''x'', ''y'' and ''z'', we obtain the trigonometric equation
		+
		+	{{Displayed math||<math>\ell\cos \gamma = a\cos \alpha -b\cos \beta\,\textrm{,}</math>}}
		+
		+	where <math>\gamma </math> is the only unknown.

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Current revision

We start by drawing three auxiliary triangles, and calling the three vertical sides x, y and z, as shown in the figure.

Using the definition of cosine, we can work out x and y from

\displaystyle \begin{align} x &= a\cos \alpha\,,\\[3pt] y &= b\cos \beta\,, \end{align}

and, for the same reason, we know that z satisfies the relation

\displaystyle z=\ell\cos \gamma\,\textrm{.}

In addition, we know that the lengths x, y and z satisfy the equality

\displaystyle z=x-y\,\textrm{.}

If we substitute in the expressions for x, y and z, we obtain the trigonometric equation

\displaystyle \ell\cos \gamma = a\cos \alpha -b\cos \beta\,\textrm{,}

where \displaystyle \gamma is the only unknown.