Precalculus

Trigonometry

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Solving General Triangles

Each triangle has \(6\) main elements (\(3\) sides and \(3\) angles). To fully specify a triangle, it is enough to know its \(3\) elements, one of which must be a side. The remaining elements can be found using the Law of Sines and Law of Cosines.

Depending on the elements given for a triangle, we'll examine \(5\) basic cases:

Case \(1.\) Three Sides \(\left( {SSS} \right)\)

Let a triangle \(ABC\) be defined by three sides \(a, b\) and \(c.\)

Case SSS: a triangle is defined by three sides a,b, and c.
Figure 1.

The unknown angles \(\alpha\) and \(\beta\) can be found using the Cosine Rule by the formulas

\[\cos \alpha = \frac{{{b^2} + {c^2} - {a^2}}}{{2bc}},\]
\[\cos \beta = \frac{{{a^2} + {c^2} - {b^2}}}{{2ac}}.\]

The third angle \(\gamma\) can also be defined by the Law of Cosines, or alternatively (if we can restore the angles \(\alpha\) and \(\beta\) knowing their cosines), by the formula

\[\gamma = 180^\circ - \alpha - \beta .\]

Case \(2.\) Two Sides and Angle Between Them \(\left( {SAS} \right)\)

In the \({SAS}\) case, we can also start with the Cosine Rule to find the third side \(c:\)

\[{c^2} = {a^2} + {b^2} - 2ab\cos \gamma .\]
Case SAS: a triangle is defined by two sides a,b, and angle gamma between them.
Figure 2.

Now we know all three sides of the triangle, so we can continue as in the \(SSS\) case. For example, we can determine the angle \(\alpha\) by the formula

\[\cos \alpha = \frac{{{b^2} + {c^2} - {a^2}}}{{2bc}},\]

and then calculate the third angle \(\beta:\)

\[\beta = 180^\circ - \alpha - \gamma .\]

Case \(3.\) Two Sides and a Non-Included Angle \(\left( {SSA} \right)\)

Consider a triangle \(ABC\) in which the sides \(b, c\) and angle \(\beta\) are known.

Case SSA: a triangle is defined by two sides b,c, and angle beta.
Figure 3.

Such a case is called the ambiguous case since several different triangles can have the given set of parameters. The unique solution is only possible when the side adjacent to the angle is shorter than or equal to the other side.

Therefore we assume that \(c \le b.\) The angle \(\gamma\) can be found from the Law of Sines:

\[\frac{b}{{\sin \beta }} = \frac{c}{{\sin \gamma }}, \Rightarrow \sin \gamma = \frac{{c\sin \beta }}{b}.\]

Note that the angle \(\gamma\) is acute (a triangle cannot have two obtuse angles). Therefore, we can uniquely find the angle \(\gamma\) knowing its sine.

The third angle \(\alpha\) can be calculated by the formula

\[\alpha = 180^\circ - \beta - \gamma .\]

The remaining side \(a\) can be found using the Sine or Cosine Rule.

Case \(4.\) One Side and Two Adjacent Angles \(\left( {ASA} \right)\)

Suppose a triangle \(ABC\) is defined by the side \(c\) and two adjacent angles \(\alpha\) and \(\beta.\)

Case ASA: a triangle is defined by side c and adjacent angles alpha, beta.
Figure 4.

We can immediately find the third angle \(\gamma:\)

\[\gamma = 180^\circ - \alpha - \beta .\]

The unknown sides \(a\) and \(b\) can be determined by the Sine Rule:

\[\frac{a}{{\sin \alpha }} = \frac{c}{{\sin \gamma }}, \Rightarrow a = \frac{{c\sin \alpha }}{{\sin \gamma }};\]
\[\frac{b}{{\sin \beta }} = \frac{c}{{\sin \gamma }}, \Rightarrow b = \frac{{c\sin \beta }}{{\sin \gamma }}.\]

Case \(5.\) Two Angles and an Non-Included Side \(\left( {AAS} \right)\)

This case is similar to the previous one. Suppose we know two angles \(\alpha,\) \(\gamma\) and the non-included side \(c.\)

Case AAS: a triangle is defined by two angles alpha and gamma, and side c.
Figure 5.

Since two angles are known, we can easily find the third angle by subtraction from \(180^\circ:\)

\[\beta = 180^\circ - \alpha - \gamma .\]

Then we use the Law of Sines again to find the unknown sides \(a\) and \(b:\)

\[\frac{a}{{\sin \alpha }} = \frac{c}{{\sin \gamma }}, \Rightarrow a = \frac{{c\sin \alpha }}{{\sin \gamma }};\]
\[\frac{b}{{\sin \beta }} = \frac{c}{{\sin \gamma }}, \Rightarrow b = \frac{{c\sin \beta }}{{\sin \gamma }}.\]

See solved problems on Page 2.

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