Example  4
In \(\triangle DEF\), \(\angle D = 25^\circ\) and \(e = 75\thinspace \rm{mm}\). Determine the range of values of \(d\) for which there is/are

  1. one oblique triangle

    In order for there to be one oblique triangle, \(a \ge b\), or in this case, \(d \ge e\).

    Therefore, \(d \ge 75\thinspace \rm{mm}\).

  2. one right triangle

    In order for there to be one right triangle, \(a = h\), or in this case,

    \(\begin{align}
     d &= h \\
     d &= e\sin D \\
     d &= 75\sin 25^\circ  \\
     d &= 31.696...\thinspace {\rm{mm}} \\
     \end{align}\)

  3. two triangles

    In order for there to be two triangles, \(b \sin A < a < b\), or in this case,

    \(\begin{align}
     e\sin D &< d < e \\
     75\sin 25^\circ &< d < 75 \\
     31.696... &< d < 75 \\
     \end{align}\)

    Therefore, \(d\) must be between \(31.696...\thinspace \rm{mm}\) and \(75\thinspace \rm{mm}\).

  4. no triangle

    In order for there to be no triangle, \(a < h\), or in this case,

    \(\begin{align}
     d &< h \\
     d &< e\sin D \\
     d &< 75\sin 25^\circ  \\
     d &< 31.696...\thinspace {\rm{mm}} \\
     \end{align}\)