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Mathematics LibreTexts

2: General Triangles

In Section 1.3 we saw how to solve a right triangle: given two sides, or one side and one acute angle, we could find the remaining sides and angles. In each case we were actually given three pieces of information, since we already knew one angle was 90◦. For a general triangle, which may or may not have a right angle, we will again need three pieces of information. The four cases are:

  • Case 1: One side and two angles
  • Case 2: Two sides and one opposite angle
  • Case 3: Two sides and the angle between them
  • Case 4: Three sides

Note that if we were given all three angles we could not determine the sides uniquely; by similarity an infinite number of triangles have the same angles. In this chapter we will learn how to solve a general triangle in all four of the above cases. Though the methods described will work for right triangles, they are mostly used to solve oblique triangles, that is, triangles which do not have a right angle. There are two types of oblique triangles: an acute triangle has all acute angles, and an obtuse triangle has one obtuse angle. As we will see, Cases 1 and 2 can be solved using the law of sines, Case 3 can be solved using either the law of cosines or the law of tangents, and Case 4 can be solved using the law of cosines.

Thumbnails: If \(C \) is acute, then \(A \) and \(B \) are also acute. Since \(A \le C \), imagine that \(A \) is in standard position in the \(xy\)-coordinate plane and that we rotate the terminal side of \(A \) counterclockwise to the terminal side of the larger angle \(C \). If we pick points \((x_{1},y_{1}) \) and \((x_{2},y_{2}) \) on the terminal sides of \(A \) and \(C \), respectively, so that their distance to the origin is the same number \(r \), then we see from the picture that \(y_{1} \le y_{2} \).