Chapter 8: Applications of Trigonometry – The Law of Sines

Applications of Trigonometry

The Law of Sines

This section explores the Law of Sines, a fundamental tool for solving oblique triangles (triangles that are not right triangles). It covers congruence axioms, the derivation and application of the Law of Sines, the ambiguous case (SSA), and formulas for the area of a triangle using trigonometric relationships.

Congruency and Oblique Triangles

• Oblique Triangle: A triangle that does not contain a right angle.

• To solve an oblique triangle, at least one side and any other two measures (angles or sides) must be known.

Congruence Axioms

• Side-Angle-Side (SAS): If two sides and the included angle of one triangle are equal to those of another triangle, the triangles are congruent.

• Angle-Side-Angle (ASA): If two angles and the included side of one triangle are equal to those of another triangle, the triangles are congruent.

• Side-Side-Side (SSS): If all three sides of one triangle are equal to those of another triangle, the triangles are congruent.

Data Required for Solving Oblique Triangles

• Case 1 (SAA or ASA): One side and two angles are known.

• Case 2 (SSA): Two sides and an angle not included between them are known. This may lead to more than one triangle (ambiguous case).

• Case 3 (SAS): Two sides and the included angle are known.

• Case 4 (SSS): All three sides are known.

Note: Knowing only the three angles (AAA) determines similarity, not congruence, so side lengths cannot be uniquely determined.

Derivation of the Law of Sines

The Law of Sines relates the sides and angles of any triangle. It is derived by dropping a perpendicular from one vertex to the opposite side, forming two right triangles within the oblique triangle.

• For triangle ABC, with sides a, b, c opposite angles A, B, C respectively:

$\sin A = \frac{h}{c}$    $\Rightarrow h = c \sin A$ $\sin C = \frac{h}{a}$    $\Rightarrow h = a \sin C$ Equating the two expressions for $h$ gives: $a \sin C = c \sin A$ $\frac{a}{\sin A} = \frac{c}{\sin C}$ Similarly, $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Law of Sines

In any triangle ABC, with sides a, b, and c opposite angles A, B, and C:

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

Using the Law of Sines

• Use the Law of Sines to solve for unknown sides or angles when given appropriate data (SAA, ASA, or SSA cases).

• Always use original given values in calculations to minimize rounding errors.

Example: Solving Triangle (SAA)

• Given: $A = 32.0^\circ$, $C = 81.8^\circ$, $a = 42.9$ cm

• Find $b$ and $c$ using the Law of Sines and the angle sum property ($A + B + C = 180^\circ$).

Example: Solving Triangle (ASA)

• Given: $C = 112.90^\circ$, $A = 31.10^\circ$, $b = 347.6$ ft

• Find $a$ using the Law of Sines after determining $B$.

Example: Real-World Application

• Given bearings and distances, use the Law of Sines to find unknown distances (e.g., distance to a fire from a ranger station).

Description of the Ambiguous Case (SSA)

When two sides and an angle opposite one of them are known (SSA), there may be zero, one, or two possible triangles.

Number of Triangles	Condition (Acute A)	Applying Law of Sines Leads to
0	$a < b$ and $\sin B > 1$	No triangle
1	$a = h$ and $a < b$	One right triangle
1	$a \geq b$	One triangle
2	$h < a < b$ and $0 < \sin B_2 < 1$	Two triangles

Number of Triangles	Condition (Obtuse A)	Applying Law of Sines Leads to
0	$a \leq b$ and $\sin B \geq 1$	No triangle
1	$a > b$ and $0 < \sin B < 1$	One triangle

Key Points for the Ambiguous Case

• If $\sin B > 1$, no triangle exists.

• If $\sin B = 1$, one right triangle exists ($B = 90^\circ$).

• If $0 < \sin B < 1$, one or two triangles may exist. If $A + B_2 < 180^\circ$, a second triangle is possible.

Examples: Ambiguous Case

• No Triangle: If calculation gives $\sin A > 1$, the triangle is not possible.

• Two Triangles: If $0 < \sin B < 1$ and $A + B_2 < 180^\circ$, solve for both possible triangles.

• One Triangle: If only one valid angle is possible, solve for the remaining sides and angles.

Area of a Triangle (SAS)

The area of a triangle can be found using two sides and the included angle:

$A = \frac{1}{2} bc \sin A$ $A = \frac{1}{2} ab \sin C$ $A = \frac{1}{2} ac \sin B$

Note: If the included angle is $90^\circ$, $\sin 90^\circ = 1$, and the formula reduces to the familiar $A = \frac{1}{2} bh$.

Example: Finding Area (SAS)

• Given: $a = 34.0$ ft, $c = 42.0$ ft, $B = 55^\circ 10'$

• Area: $A = \frac{1}{2} (34.0)(42.0) \sin 55^\circ 10' \approx 586$ ft2

Summary Table: Law of Sines and Ambiguous Case

Case	Given	Possible Solutions
SAA/ASA	1 side, 2 angles	Unique solution
SSA	2 sides, angle opposite one	0, 1, or 2 solutions (ambiguous case)
SAS	2 sides, included angle	Unique solution
SSS	3 sides	Unique solution

Additional info: The Law of Sines is essential for solving real-world problems involving non-right triangles, such as navigation, surveying, and physics applications.