The Law of Sines and Its Applications in Solving Oblique Triangles

9.2 The Law of Sines

Introduction to the Law of Sines

The Law of Sines is a fundamental trigonometric relationship used to solve oblique (non-right) triangles. It relates the lengths of the sides of a triangle to the sines of its angles. If $A$, $B$, and $C$ are the measures of the angles of any triangle, and $a$, $b$, and $c$ are the lengths of the sides opposite these angles, then:

$\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Cases for Applying the Law of Sines

The Law of Sines can be used to solve triangles when certain combinations of sides and angles are known. The following table summarizes the cases:

Triangle	Description of Case	Abbreviation of Case
	Side-Angle-Angle: two angles and any side are known	SAA
	Angle-Side-Angle: two angles and the side between them are known	ASA
	Side-Side-Angle: two sides and an angle not between them are known	SSA
	Side-Side-Side: all three sides are known	SSS
	Angle-Angle-Angle: all three angles are known	AAA

Note: The Law of Sines is most useful for solving the SAA, ASA, and SSA cases. The SSS and AAA cases require other methods or do not determine a unique triangle.

Solving the SAA and ASA Cases

When two angles and any side of an oblique triangle are known, the Law of Sines can be used to find the remaining sides. The process involves:

• Using the Law of Sines to set up proportions between known and unknown sides and angles.

• Solving for the unknown side(s) using algebraic manipulation.

• Rounding answers to two decimal places unless otherwise specified.

Example: Given $A = 40^\circ$, $B = 60^\circ$, and $a = 10$, find $b$.

$\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B}$ $\displaystyle \frac{10}{\sin 40^\circ} = \frac{b}{\sin 60^\circ}$ $\displaystyle b = \frac{10 \cdot \sin 60^\circ}{\sin 40^\circ}$

Solving the SSA Case (The Ambiguous Case)

The SSA case is known as the Ambiguous Case because it can result in zero, one, or two possible triangles depending on the given values. The following table summarizes the possibilities:

Value of $\frac{b}{\sin B}$	Number of Triangles	Possible Triangles	Description
$\sin B > 1$	No triangle		No angle exists; side is too short to reach the opposite side.
$\sin B = 1$	One right triangle		The measure of $B$ is $90^\circ$.
$0 < \sin B < 1$	One oblique triangle		If there is one solution for $B$, the triangle is oblique with either one acute or one obtuse angle.
$0 < \sin B < 1$ (with two solutions)	Two oblique triangles		If there are two solutions for $B$, there are two possible triangles: one with two acute angles and one with one obtuse angle.

Summary of Cases for the Law of Sines

The Law of Sines can be used to solve the following cases for oblique triangles:

• SAA Case: Two angles and any side are known.

• ASA Case: Two angles and the included side are known.

• SSA Case (Ambiguous Case): Two sides and a non-included angle are known.

Each case is illustrated below:

• The SAA Case

• The ASA Case

• The SSA Case (Ambiguous Case)

Applied Problems Involving Oblique Triangles

The Law of Sines is frequently used to solve real-world problems involving oblique triangles, such as navigation, surveying, and physics. The key steps are:

• Identify the known sides and angles.

• Determine which case applies (SAA, ASA, or SSA).

• Apply the Law of Sines to find unknown sides or angles.

• Interpret the solution in the context of the problem.

Additional info: The notes provide a comprehensive overview of the Law of Sines, including its formula, applicable cases, and the ambiguous case analysis. The tables have been recreated in HTML format, and examples have been expanded for clarity.