Solving Non-Right-Angled Triangles

Introduction

Key Concepts

The Basics of Triangle Types

Triangles can be classified based on their angles and sides. Understanding these classifications is crucial when applying trigonometric laws:

• By Angles:
  • Acute Triangle: All angles are less than 90°.
  • Obtuse Triangle: One angle is greater than 90°.
  • Right Triangle: One angle is exactly 90°.
• By Sides:
  • Equilateral Triangle: All three sides are equal.
  • Isosceles Triangle: Two sides are equal.
  • Scalene Triangle: All sides are of different lengths.

The Law of Sines

The Law of Sines relates the sides of a triangle to the sines of its opposite angles. It is particularly useful in solving triangles that are not right-angled.

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

Where:

• a, b, c are the lengths of the sides opposite angles A, B, C respectively.

Applications:

• ASA (Angle-Side-Angle): Two angles and the included side are known.
• AAS (Angle-Angle-Side): Two angles and a non-included side are known.
• SSA (Side-Side-Angle): Two sides and a non-included angle are known.

The Law of Cosines

The Law of Cosines generalizes the Pythagorean theorem for any triangle, regardless of its angles.

$$c^2 = a^2 + b^2 - 2ab \cos C$$

Similarly, the law can be written for the other sides:

$$a^2 = b^2 + c^2 - 2bc \cos A$$ $$b^2 = a^2 + c^2 - 2ac \cos B$$

Applications:

• SAS (Side-Angle-Side): Two sides and the included angle are known.
• SSS (Side-Side-Side): All three sides are known.

Solving Triangles Using the Laws of Sines and Cosines

Solving a triangle involves finding all unknown sides and angles given a set of known parameters. The choice between the Law of Sines and the Law of Cosines depends on the available information.

• Case 1: SSS (Side-Side-Side)
• Calculate one angle using the Law of Cosines.
• Use the Law of Sines to find the remaining angles.
• Case 2: SAS (Side-Angle-Side)
• Use the Law of Cosines to find the third side.
• Apply the Law of Sines to determine the remaining angles.
• Case 3: ASA or AAS (Angle-Side-Angle or Angle-Angle-Side)
• Use the fact that the sum of angles in a triangle is 180° to find the third angle.
• Employ the Law of Sines to find the unknown sides.
• Case 4: SSA (Side-Side-Angle)
• Use the Law of Sines to find the possible solutions for the unknown angles.
• Be cautious of the ambiguous case where two different triangles may satisfy the given conditions.

The Ambiguous Case in the Law of Sines

The SSA (Side-Side-Angle) configuration can lead to zero, one, or two possible triangles:

• No Solution: If the side opposite the given angle is shorter than the altitude, no triangle exists.
• One Solution: If the side opposite the given angle is equal to the altitude or longer than it.
• Two Solutions: If the side opposite the given angle is longer than the altitude but shorter than the given side adjacent to the angle.

Trigonometric Identities and Relationships

Understanding basic trigonometric identities enhances the ability to manipulate and solve equations involving the Laws of Sines and Cosines. Key identities include:

• Sum of Angles: $A + B + C = 180°$
• Extended Law of Sines: $a = 2R \sin A$, $b = 2R \sin B$, $c = 2R \sin C$, where $R$ is the circumradius.
• Area Formulas:
  • $Area = \frac{1}{2}ab \sin C$
  • $Area = \frac{a^2 \sin B \sin C}{2 \sin A}$

Application Examples

Let's consider an example to illustrate the application of these laws.

Example 1: In triangle ABC, side $a = 7$, side $b = 10$, and angle $A = 45°$. Find side $c$ and angles $B$ and $C$.

• Step 1: Use the Law of Cosines to find side $c$: $$c^2 = a^2 + b^2 - 2ab \cos C$$ However, since angle $A$ is given, it's more straightforward to find another angle first using the Law of Sines.
• Step 2: Apply the Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B}$$ $$\frac{7}{\sin 45°} = \frac{10}{\sin B}$$ $$\sin B = \frac{10 \sin 45°}{7} \approx \frac{10 \times 0.7071}{7} \approx 1.0102$$ Since $\sin B$ cannot exceed 1, this indicates no solution, implying a mistake in the setup or the need to reassess the problem data.

Note: Always verify the feasibility of solutions, especially in SSA cases, to account for the ambiguous scenarios.

Advanced Concepts

Derivation of the Law of Sines and Cosines

Understanding the derivation of these laws provides deeper insights into their applications and limitations.

Derivation of the Law of Sines

Consider triangle ABC with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$ respectively. Drop a perpendicular from vertex A to side $a$, forming height $h$.

Using trigonometric relationships: $$\sin B = \frac{h}{a} \Rightarrow h = a \sin B$$ Similarly, $$\sin C = \frac{h}{b} \Rightarrow h = b \sin C$$ Setting the two expressions for $h$ equal: $$a \sin B = b \sin C$$ Thus, $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ This is the Law of Sines.

Derivation of the Law of Cosines

Using the same triangle, apply the Pythagorean theorem to the height $h$:

$$c^2 = b^2 + h^2$$ Substituting $h = a \sin B$: $$c^2 = b^2 + a^2 \sin^2 B$$ Using the identity $\sin^2 B = 1 - \cos^2 B$: $$c^2 = b^2 + a^2 (1 - \cos^2 B)$$ $$c^2 = a^2 + b^2 - a^2 \cos^2 B$$ To find a relationship involving $\cos C$, a more comprehensive derivation using vector components or coordinate geometry can be employed, leading to the generalized Law of Cosines: $$c^2 = a^2 + b^2 - 2ab \cos C$$

Solving Triangles Using Systems of Equations

Advanced triangle solving often involves setting up and solving systems of equations derived from the Laws of Sines and Cosines.

Example 2: Given triangle ABC with sides $a = 8$, $b = 15$, and angle $A = 60°$, find side $c$ and angles $B$ and $C$.

• Step 1: Apply the Law of Cosines to find side $c$: $$c^2 = a^2 + b^2 - 2ab \cos C$$ However, since angle $A$ is given, it's more efficient to first find angle $B$ using the Law of Sines.
• Step 2: Use the Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ $$\frac{8}{\sin 60°} = \frac{15}{\sin B}$$ $$\sin B = \frac{15 \sin 60°}{8} = \frac{15 \times \frac{\sqrt{3}}{2}}{8} = \frac{15\sqrt{3}}{16} \approx 1.623$$ Since $\sin B > 1$, no such triangle exists, indicating the initial assumptions or data require review.

Note: Carefully analyze the given data to ensure valid triangle configurations, especially when dealing with SSA scenarios that may lead to ambiguous or impossible cases.

Understanding the Ambiguous Case Through Analytical Geometry

The ambiguous case arises in SSA configurations where two different triangles satisfy the given conditions. An analytical approach using geometry can clarify the number of possible solutions.

Given: Two sides $a$, $b$, and a non-included angle $A$.

Conditions:

• If $a < b \sin A$, no solution exists.
• If $a = b \sin A$, exactly one right-angled triangle exists.
• If $b \sin A < a < b$, two distinct triangles are possible.
• If $a \geq b$, exactly one triangle exists.

These conditions stem from the geometric possibilities of extending or shortening sides based on the given angle and side lengths.

Interdisciplinary Connections: Engineering and Physics

The ability to solve non-right-angled triangles is indispensable in fields like engineering and physics. For instance:

• Engineering: Determining forces in non-parallel directions requires solving triangles to find resultant vectors.
• Physics: Analyzing projectile motion or resolving vectors in equilibrium involves trigonometric solutions of non-right-angled triangles.
• Geography: Triangulation methods for mapping and surveying rely on these trigonometric principles.

Proof of the Law of Cosines Using Vector Algebra

Vector algebra offers a robust framework for proving the Law of Cosines:

Consider vectors $\vec{A}$ and $\vec{B}$ representing sides $b$ and $c$ of triangle ABC, respectively. The side $a$ is the difference vector:

$$\vec{a} = \vec{B} - \vec{A}$$

Taking the magnitude squared of both sides: $$a^2 = (\vec{B} - \vec{A}) \cdot (\vec{B} - \vec{A}) = \vec{B} \cdot \vec{B} - 2 \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{A}$$ $$a^2 = b^2 + c^2 - 2bc \cos \theta$$

Where $\theta$ is the angle between vectors $\vec{A}$ and $\vec{B}$, corresponding to angle $C$ in triangle ABC. This derivation confirms the Law of Cosines.

Inverse Problems and Real-World Applications

Inverse problems involve determining unknown parameters from known outcomes, a common scenario in real-world applications.

Example 3: A surveyor measures two sides of a triangular plot of land and the included angle. Using the Law of Cosines, the surveyor can determine the third side, facilitating land division and construction planning.

Similarly, in navigation, determining the shortest path or calculating bearings involves solving non-right-angled triangles to ensure accurate course plotting.

Numerical Methods for Solving Complex Triangles

In cases where analytical solutions are cumbersome, numerical methods like the Newton-Raphson technique can approximate the solutions to non-right-angled triangles with high precision.

These methods are particularly useful in computational applications where multiple iterations refine the approximation of sides and angles, enhancing efficiency and accuracy.

Comparison Table

Aspect	Law of Sines	Law of Cosines
Purpose	Relates sides to sines of opposite angles.	Generalizes Pythagorean theorem for any triangle.
Applicable Cases	ASA, AAS, SSA.	SAS, SSS.
Equations	$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$	$c^2 = a^2 + b^2 - 2ab \cos C$
Advantages	Simple to apply in many cases; useful for finding angles and sides.	Essential for solving triangles where Law of Sines is insufficient; handles SSS and SAS cases.
Limitations	Ambiguous when applied to SSA; may yield two solutions.	More complex calculations; primarily used when two sides and an included angle are known or when all three sides are known.

Summary and Key Takeaways