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Solution 4.2:8

From Förberedande kurs i matematik 1

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We start by drawing three auxiliary triangles, and calling the three vertical sides ''x'', ''y'' and ''z'', as shown in the figure.
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<center> [[Bild:4_2_8-1(2).gif]] </center>
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<center> [[Bild:4_2_8-2(2).gif]] </center>
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[[Image:4_2_8.gif|center]]
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[[Bild:4_2_8.gif|center]]
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Using the definition of cosine, we can work out ''x'' and ''y'' from
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{{Displayed math||<math>\begin{align}
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x &= a\cos \alpha\,,\\[3pt]
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y &= b\cos \beta\,,
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\end{align}</math>}}
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and, for the same reason, we know that ''z'' satisfies the relation
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{{Displayed math||<math>z=\ell\cos \gamma\,\textrm{.}</math>}}
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In addition, we know that the lengths ''x'', ''y'' and ''z'' satisfy the equality
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{{Displayed math||<math>z=x-y\,\textrm{.}</math>}}
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If we substitute in the expressions for ''x'', ''y'' and ''z'', we obtain the trigonometric equation
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{{Displayed math||<math>\ell\cos \gamma = a\cos \alpha -b\cos \beta\,\textrm{,}</math>}}
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where <math>\gamma </math> is the only unknown.

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

xy=acos=bcos

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

z=cos.

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

z=xy.

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

cos=acosbcos,

where is the only unknown.