Solution 4.2:8

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Current revision (12:23, 9 October 2008) (edit) (undo)
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We start by drawing three auxiliary triangles, and calling the three vertical sides
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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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<math>x,\ y</math>
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and
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<math>z</math>, as shown in the figure.
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[[Image:4_2_8.gif|center]]
[[Image:4_2_8.gif|center]]
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Using the definition of cosine, we can work out
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Using the definition of cosine, we can work out ''x'' and ''y'' from
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<math>x\text{ }</math>
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and
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<math>y</math>
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from
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<math>x=a\cos \alpha </math>
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<math>y=b\cos \beta </math>
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and, for the same reason, we know that
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<math>z\text{ }</math>
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satisfies the relation
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<math>z=l\cos \gamma </math>
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In addition, we know that the lengths
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<math>x,\ y</math>
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and
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<math>z</math>
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satisfy the equality
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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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<math>z=x-y</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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If we substitute in the expressions for
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In addition, we know that the lengths ''x'', ''y'' and ''z'' satisfy the equality
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<math>x,\ y</math>
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and
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<math>z</math>, we obtain the trigonometric equation
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{{Displayed math||<math>z=x-y\,\textrm{.}</math>}}
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<math>l\cos \gamma =a\cos \alpha -b\cos \beta </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
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where <math>\gamma </math> is the only unknown.
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<math>\gamma </math>
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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.

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