Using the Sine Rule and Cosine Rule for Solving Triangle Problems

Introduction

Key Concepts

The Sine Rule

The sine rule, also known as the law of sines, establishes a relationship between the lengths of the sides of a triangle and the sines of its opposite angles. It is particularly useful in solving triangles that are not right-angled.

The sine rule is expressed as:

Where:

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

**Applications of the Sine Rule:**

• Solving for Unknown Sides: Given two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA), the sine rule can be used to find the lengths of the remaining sides.
• Determining Unknown Angles: When given two sides and a non-included angle, the sine rule helps in finding the other angles.

**Example Problem:** Find the length of side b in triangle ABC where angle A = 30°, angle B = 45°, and side a = 10 units.

**Solution:** First, find angle C: $$ C = 180° - A - B = 180° - 30° - 45° = 105° $$ Using the sine rule: $$ \frac{a}{\sin A} = \frac{b}{\sin B} $$ $$ \frac{10}{\sin 30°} = \frac{b}{\sin 45°} $$ $$ \frac{10}{0.5} = \frac{b}{0.7071} $$ $$ 20 = \frac{b}{0.7071} $$ $$ b = 20 \times 0.7071 \approx 14.14 \text{ units} $$

The Cosine Rule

The cosine rule, or law of cosines, relates the lengths of all three sides of a triangle to the cosine of one of its angles. It is particularly useful when dealing with triangles where the sine rule is cumbersome or inapplicable, such as when dealing with the side-side-side (SSS) or side-angle-side (SAS) scenarios.

The cosine rule is given by:

Where:

• a, b, c are the lengths of the sides of the triangle.
• C is the measure of the angle opposite side c.

**Applications of the Cosine Rule:**

• Solving for an Unknown Side: Given two sides and the included angle (SAS), the cosine rule can determine the length of the third side.
• Finding an Unknown Angle: When all three sides are known (SSS), the cosine rule can be used to find any of the angles.

**Example Problem:** Find the measure of angle C in triangle ABC where sides a = 7 units, b = 10 units, and c = 5 units.

**Solution:** Using the cosine rule: $$ c^2 = a^2 + b^2 - 2ab \cos C $$ $$ 5^2 = 7^2 + 10^2 - 2 \times 7 \times 10 \cos C $$ $$ 25 = 49 + 100 - 140 \cos C $$ $$ 25 = 149 - 140 \cos C $$ $$ -124 = -140 \cos C $$ $$ \cos C = \frac{124}{140} = 0.8857 $$ $$ C = \cos^{-1}(0.8857) \approx 27.0° $$

Understanding Triangles

Triangles are classified based on their angles and sides. Understanding the properties and types of triangles is crucial for applying the sine and cosine rules effectively.

• Acute Triangle: All angles are less than 90°.
• Obtuse Triangle: One angle is greater than 90°.
• Scalene Triangle: All sides and angles are different.
• Isosceles Triangle: Two sides and two angles are equal.

Steps to Solve Triangle Problems Using Sine and Cosine Rules

1. Identify Known Quantities: Determine which sides and angles are known.
2. Decide Which Rule to Apply: Use the sine rule for AAS, ASA, or SSA cases. Use the cosine rule for SAS or SSS cases.
3. Set Up Equations: Apply the appropriate rule to set up an equation relating the known and unknown quantities.
4. Solve for Unknowns: Solve the equation(s) to find the unknown sides or angles.
5. Verify Solutions: Check if the solutions satisfy the original triangle properties.

Common Pitfalls and Tips

• Ambiguous Case: In SSA scenarios, there might be two possible solutions. Always consider the possibility of multiple triangles.
• Rounding Errors: Be cautious with rounding during intermediate steps to avoid inaccuracies.
• Angle Sum Property: Always ensure that the sum of angles in a triangle equals 180°.

Advanced Concepts

Derivation of the Sine Rule

The sine rule can be derived using the properties of triangles and basic trigonometric identities. Consider a triangle ABC with sides a, b, and c opposite angles A, B, and C, respectively.

By dropping a perpendicular from vertex C to side AB, we create a right-angled triangle where the height (h) can be expressed as:

Since both expressions equal the height, we can set them equal to each other:

Rearranging terms gives us the sine rule:

Extending this to include the third side and angle, the complete sine rule is established as:

Derivation of the Cosine Rule

The cosine rule is derived using the Pythagorean theorem extended to non-right-angled triangles. Consider triangle ABC with side lengths a, b, and c.

Using the Law of Cosines for side c:

This formula allows the calculation of the third side when two sides and the included angle are known, or conversely, the calculation of the included angle when all three sides are known.

Inverse Sine and Cosine Functions

Inverse functions are essential when solving for angles after applying the sine or cosine rules. They allow the determination of angle measures from their sine or cosine values.

For example, if $\sin C = 0.5$, then:

Similarly, if $\cos C = 0.866$, then:

Handling the Ambiguous Case (SSA)

When applying the sine rule in SSA scenarios, there can be one, two, or no possible solutions. This depends on the given sides and angles:

• No Solution: If the height drawn from the known angle is greater than the given side opposite another angle.
• One Solution: If the height is equal to or greater than the given side opposite another angle.
• Two Solutions: If the given side is longer than the height but less than the sum of the height and the adjacent side.

**Example Problem:** Given triangle ABC with angle A = 30°, side a = 10 units, and side b = 15 units, determine the possible measures of angle B.

**Solution:** Using the sine rule: $$ \frac{a}{\sin A} = \frac{b}{\sin B} $$ $$ \frac{10}{\sin 30°} = \frac{15}{\sin B} $$ $$ \frac{10}{0.5} = \frac{15}{\sin B} $$ $$ 20 = \frac{15}{\sin B} $$ $$ \sin B = \frac{15}{20} = 0.75 $$ $$ B = \sin^{-1}(0.75) \approx 48.6° $$

However, since $\sin(180° - θ) = \sin θ$, there is a second possible angle:

Thus, there are two possible triangles with angles B ≈ 48.6° and B' ≈ 131.4°.

Applications in Real-World Scenarios

The sine and cosine rules are not just theoretical constructs; they have practical applications in various fields:

• Navigation and Surveying: Determining unknown distances and angles for mapping and navigation purposes.
• Engineering: Designing structures where non-right-angled triangles are involved.
• Astronomy: Calculating distances and angles between celestial bodies.
• Physics: Resolving vector components in mechanics.

Interdisciplinary Connections

Trigonometric principles, specifically the sine and cosine rules, bridge mathematics with other disciplines:

• Physics: Understanding forces, motion, and wave properties.
• Engineering: Structural analysis, electrical circuits, and signal processing.
• Computer Graphics: Rendering angles and distances in digital environments.
• Economics: Modeling cyclical patterns and oscillations.

Challenging Problems and Solutions

To deepen understanding, tackle more complex problems that integrate multiple concepts:

**Problem:** In triangle ABC, angle A = 45°, side a = 7 units, and side b = 10 units. Find the length of side c and all the angles.

**Solution:** First, use the sine rule to find angle B: $$ \frac{a}{\sin A} = \frac{b}{\sin B} $$ $$ \frac{7}{\sin 45°} = \frac{10}{\sin B} $$ $$ \frac{7}{0.7071} = \frac{10}{\sin B} $$ $$ 9.8995 = \frac{10}{\sin B} $$ $$ \sin B = \frac{10}{9.8995} \approx 1.0102 $$

Since the sine of an angle cannot exceed 1, there is no solution in this case, indicating that the given dimensions do not form a valid triangle.

**Alternative Problem:** Given triangle DEF with sides d = 8 units, e = 6 units, and angle D = 60°, find side f and angles E and F.

**Solution:** First, use the cosine rule to find side f: $$ f^2 = d^2 + e^2 - 2de \cos D $$ $$ f^2 = 8^2 + 6^2 - 2 \times 8 \times 6 \cos 60° $$ $$ f^2 = 64 + 36 - 96 \times 0.5 $$ $$ f^2 = 100 - 48 = 52 $$ $$ f = \sqrt{52} \approx 7.21 \text{ units} $$ Next, use the sine rule to find angle E: $$ \frac{e}{\sin E} = \frac{d}{\sin D} $$ $$ \frac{6}{\sin E} = \frac{8}{\sin 60°} $$ $$ \frac{6}{\sin E} = \frac{8}{0.8660} $$ $$ \frac{6}{\sin E} \approx 9.2376 $$ $$ \sin E = \frac{6}{9.2376} \approx 0.6495 $$ $$ E = \sin^{-1}(0.6495) \approx 40.4° $$ Finally, find angle F: $$ F = 180° - D - E = 180° - 60° - 40.4° = 79.6° $$

Comparison Table

Summary and Key Takeaways