Using the Law of Sines to Solve Oblique Triangles and Find Their Area

Oblique triangles are triangles that don’t have a right angle. Because the familiar SOH CAH TOA relationships apply only to right‑angled triangles, they cannot be used directly on oblique triangles. For these, all the SOHCAHTOA stuff we learned won’t work anymore. Instead, we rely on the Law of Sines, a relationship that works for any non‑right triangle.

The Law of Sines Formula

If the three angles of an oblique triangle are labeled A, B, C and the sides opposite them are a, b, c, the Law of Sines states

[ \frac{a}{\sin A}= \frac{b}{\sin B}= \frac{c}{\sin C}. ]

Professor Dave emphasizes this as “the relationship between all of these will be A over sine A equals B over sine B equals C over sine C.” The equality lets us swap known side‑angle pairs for unknown ones.

Applying the Law of Sines

Side‑Angle‑Angle (SAA)

When two angles are known, the third angle follows from the triangle‑sum rule:

[ \text{third angle}=180^\circ-(\text{first angle}+\text{second angle}). ]

With one side‑angle pair, the ratio (a/\sin A) (or its equivalent) can be used to solve for the remaining two sides. Calculations typically involve degrees and a calculator for approximate side lengths.

Angle‑Side‑Angle (ASA)

If the known side lies between the two known angles, we first find the third angle by subtracting the two given angles from 180°. Then we plug the values into the Law of Sines to obtain the unknown side(s).

Side‑Side‑Angle (SSA)

The SSA case is the most delicate. First we find the angle opposite a known side using

[ \sin(\text{unknown angle})=\frac{\text{known side}\times\sin(\text{known angle})}{\text{other known side}}. ]

If the computed sine exceeds 1, the law of sines will prove that the third side can’t possibly connect the other two, and the triangle can’t exist. When the sine value is valid, two situations can arise:

• No solution – the sine value is greater than 1, which is impossible for the sine function.
• Ambiguous case – a valid sine value may correspond to two different angles, so the SSA triangle can have two distinct solutions.

Area of an Oblique Triangle

The familiar area formula for any triangle is

[ \text{Area}= \frac{1}{2}\times\text{base}\times\text{height}. ]

For oblique triangles we often do not know the height directly, so an alternative formula is useful. Using two sides and the included angle, the area can be expressed as

[ \text{Area}= \frac{1}{2}\,a\,b\,\sin C, ]

and similarly for the other side‑pair combinations. Professor Dave notes, “the area is equal to the product of the lengths of two sides and the sine of the angle between them.” This provides three interchangeable versions, one for each pair of sides and its included angle.

Conclusion

The Law of Sines offers a systematic way to solve oblique triangles in the SAA, ASA, and SSA configurations, while also revealing when a set of measurements cannot form a triangle. Additionally, the sine‑based area formula lets us compute the area without explicitly finding a height, completing the toolkit for handling any non‑right triangle.