Solving Non-Right-Angled Triangles

Introduction

Key Concepts

Understanding Non-Right-Angled Triangles

A non-right-angled triangle, also known as an oblique triangle, lacks a 90-degree angle. Unlike right-angled triangles, solving these triangles requires different approaches since the Pythagorean theorem is not directly applicable. Instead, the Laws of Sines and Cosines provide the necessary tools to find unknown sides and angles.

The Law of Sines

The Law of Sines relates the ratios of the lengths of a triangle's sides to the sines of its opposite angles. It is particularly useful in solving acute and obtuse triangles where two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA) are known.

The Law of Sines is expressed as: $$ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} $$ where \( a, b, c \) are the lengths of the sides opposite angles \( A, B, C \) respectively.

Example: Given a triangle with sides \( a = 7 \), angle \( A = 30^\circ \), and angle \( B = 45^\circ \), find side \( b \).

Using the Law of Sines: $$ \frac{7}{\sin(30^\circ)} = \frac{b}{\sin(45^\circ)} $$ $$ \frac{7}{0.5} = \frac{b}{0.7071} $$ $$ 14 = \frac{b}{0.7071} \Rightarrow b \approx 9.899 $$

The Law of Cosines

The Law of Cosines generalizes the Pythagorean theorem for any triangle, allowing the calculation of a side when two sides and the included angle (SAS) are known, or an angle when all three sides (SSS) are known.

The Law of Cosines is given by: $$ c^2 = a^2 + b^2 - 2ab\cos(C) $$ Similarly, this applies cyclically for other sides and angles.

Example: In a triangle with sides \( a = 5 \), \( b = 7 \), and angle \( C = 60^\circ \), find side \( c \).

Applying the Law of Cosines: $$ c^2 = 5^2 + 7^2 - 2 \times 5 \times 7 \cos(60^\circ) $$ $$ c^2 = 25 + 49 - 70 \times 0.5 $$ $$ c^2 = 74 - 35 = 39 \Rightarrow c \approx 6.245 $$

Solving Triangles Using the Law of Sines

To solve a triangle using the Law of Sines, follow these steps:

1. Identify the known sides and angles.
2. Set up the ratio using the Law of Sines.
3. Solve for the unknown sides or angles.
4. Check for the Ambiguous Case in SSA triangles.
5. Verify your solutions.

Ambiguous Case: In SSA scenarios, there may be two possible solutions, one solution, or no solution, depending on the given values. It's crucial to consider this when solving for angles.

Solving Triangles Using the Law of Cosines

The Law of Cosines is particularly effective in the following cases:

• SAS (Side-Angle-Side): When two sides and the included angle are known, use the Law of Cosines to find the third side.
• SSS (Side-Side-Side): When all three sides are known, use the Law of Cosines to find one angle, then apply the Law of Sines or Cosines to find the remaining angles.

Example (SSS): Given a triangle with sides \( a = 8 \), \( b = 15 \), and \( c = 17 \), find angle \( C \).

Applying the Law of Cosines: $$ 17^2 = 8^2 + 15^2 - 2 \times 8 \times 15 \cos(C) $$ $$ 289 = 64 + 225 - 240 \cos(C) $$ $$ 289 = 289 - 240 \cos(C) \Rightarrow 240 \cos(C) = 0 \Rightarrow \cos(C) = 0 \Rightarrow C = 90^\circ $$

Combining the Laws of Sines and Cosines

Often, solving a non-right-angled triangle requires a combination of both laws. For instance, after finding one angle using the Law of Cosines, the Law of Sines can efficiently find the remaining sides and angles.

Comprehensive Example: Given a triangle with sides \( a = 10 \), \( b = 7 \), and angle \( A = 45^\circ \), find sides \( c \) and angle \( B \).

1. Apply the Law of Sines to find angle \( B \): $$ \frac{10}{\sin(45^\circ)} = \frac{7}{\sin(B)} $$ $$ \frac{10}{0.7071} = \frac{7}{\sin(B)} \Rightarrow \sin(B) = \frac{7 \times 0.7071}{10} \approx 0.49497 \Rightarrow B \approx 29.7^\circ $$ 2. Find angle \( C \): $$ C = 180^\circ - A - B = 180^\circ - 45^\circ - 29.7^\circ = 105.3^\circ $$ 3. Use the Law of Sines to find side \( c \): $$ \frac{c}{\sin(105.3^\circ)} = \frac{10}{\sin(45^\circ)} \Rightarrow c = \frac{10 \times \sin(105.3^\circ)}{\sin(45^\circ)} \approx \frac{10 \times 0.966}{0.7071} \approx 13.7 $$

Practical Applications

Understanding how to solve non-right-angled triangles is vital in various fields such as engineering, physics, astronomy, and architecture. It facilitates the determination of distances and angles in real-world scenarios where right angles are not present.

Comparison Table

Summary and Key Takeaways