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 | + | 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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[[Image:4_2_8.gif|center]] | [[Image:4_2_8.gif|center]] | ||
- | Using the definition of cosine, we can work out | + | Using the definition of cosine, we can work out ''x'' and ''y'' from |
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+ | {{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 | |
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+ | {{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 | + | where <math>\gamma </math> is the only unknown. |
- | <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
\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.