In this section, we will discuss the Law of SINES and its application in solving oblique triangles. Up until now, the triangles we have solved have all been right triangles. However, we can use sine and cosine to solve oblique triangles, which are triangles WITHOUT a right angle.
Solving Oblique Triangles
To solve an oblique triangle, you must know the measure of at least one SIDE, and any two other parts of the triangle. The possibilities are:
1) AAS
2) ASA
3) SSA
4) SAS
5) SSS
Only three of these situations can be solved with the Law of Sines, and the other two will use the Law of Cosines.
Law of Sines Formula
The Law of Sines states that for any triangle ABC with sides a, b, and c, the following formula can be used:
[ \frac{b}{\sin B} = \frac{a}{\sin A} = \frac{c}{\sin C} ]
Reciprocal Form
The Law of Sines can also be written in reciprocal form:
[ \frac{sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c} ]
This formula is applicable when angle A is acute or obtuse.
The AAS Case
For the AAS case, we can use the Law of Sines to find the remaining sides and angles in the triangles.
The ASA Case
Similarly, for the ASA case, the Law of Sines can be applied to find the missing parts of the triangle.
The SSA Case (Ambiguous Case)
The SSA case, also known as the ambiguous case, raises a unique challenge in solving triangles. Depending on the information given, it may result in 0, 1, or 2 possible triangles. This ambiguity arises from the nature of the SSA arrangement in the triangle.
Ambiguity in SSA Case
When given two sides and the NON-included angle (SSA), multiple triangles can be constructed. The ambiguity is resolved based on the range of values for side "a".
Applications
The law of Sines can be applied to real-world scenarios. For example, in determining the height of a telephone pole or the altitude of a hot-air balloon. The Law of Sines also has applications in finding the area of oblique triangles.
Area of an Oblique Triangle
The area of any triangle can be calculated using the formula:
[ \frac{1}{2} \times a \times h ]
Where h can be expressed in terms of the sides a, b, and c.
In conclusion, the Law of Sines provides a valuable tool for solving oblique triangles and has various real-world applications. It offers a versatile approach to solving triangles and is particularly effective in scenarios involving non-right angles.
