Problem Solving with Non-Right Triangles

Introduction

Key Concepts

The Basics of Non-Right Triangles

A non-right triangle is any triangle that does not contain a right angle (90 degrees). These triangles can be categorized as acute (all angles less than 90 degrees), obtuse (one angle greater than 90 degrees), or scalene and isosceles based on their side lengths. Solving non-right triangles involves finding unknown sides or angles using trigonometric principles.

The Sine Rule

The Sine Rule, also known as the Law of Sines, establishes a relationship between the lengths of a triangle's sides and the sines of its opposite angles. It is particularly useful in solving triangles that do not contain a right angle.

The Sine Rule is expressed as: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

• Where:
  • a, b, c: Lengths of the sides of the triangle.
  • A, B, C: Measures of the angles opposite the respective sides.

Applications: The Sine Rule is typically used in two scenarios:

1. ASA (Angle-Side-Angle): Two angles and any side of the triangle are known.
2. AAS (Angle-Angle-Side): Two angles and a non-included side of the triangle are known.

Example: Suppose we have a triangle with angles A = 30°, B = 45°, and side a = 10 units. To find side b:

Using the Sine Rule: $$\frac{10}{\sin 30°} = \frac{b}{\sin 45°}$$ $$\frac{10}{0.5} = \frac{b}{0.7071}$$ $$20 = \frac{b}{0.7071}$$ $$b = 20 \times 0.7071$$ $$b \approx 14.14 \text{ units}$$

The Cosine Rule

The Cosine Rule, or Law of Cosines, connects the lengths of a triangle's sides with the cosine of one of its angles. This rule is particularly useful in solving triangles when certain combinations of sides and angles are known.

The Cosine Rule is given by: $$c^2 = a^2 + b^2 - 2ab \cos C$$

• Where:
  • a, b, c: Lengths of the sides of the triangle.
  • C: Measure of the angle opposite side c.

Applications: The Cosine Rule is applied in two main scenarios:

1. SAS (Side-Angle-Side): Two sides and the included angle are known.
2. SSS (Side-Side-Side): All three sides of the triangle are known.

Example: Consider a triangle with sides a = 7 units, b = 10 units, and angle C = 60°. To find side c:

Using the Cosine Rule: $$c^2 = 7^2 + 10^2 - 2 \times 7 \times 10 \times \cos 60°$$ $$c^2 = 49 + 100 - 140 \times 0.5$$ $$c^2 = 149 - 70$$ $$c^2 = 79$$ $$c = \sqrt{79}$$ $$c \approx 8.89 \text{ units}$$

Area of a Triangle Using Trigonometry

Calculating the area of a non-right triangle can be efficiently done using trigonometric formulas. One such method employs the formula involving two sides and the sine of the included angle.

The area Δ is given by: $$Δ = \frac{1}{2} a b \sin C$$

• Where:
  • a, b: Lengths of two sides.
  • C: Measure of the included angle.

Example: Given a triangle with sides a = 5 units, b = 7 units, and included angle C = 45°, the area is calculated as:

$$Δ = \frac{1}{2} \times 5 \times 7 \times \sin 45°$$ $$Δ = \frac{1}{2} \times 35 \times 0.7071$$ $$Δ = 17.5 \times 0.7071$$ $$Δ \approx 12.37 \text{ square units}$$

Solving Triangles Using the Sine and Cosine Rules

To solve a non-right triangle means to find all its unknown sides and angles. Depending on the known information, either the Sine Rule or the Cosine Rule can be applied, or a combination of both.

Steps to Solve a Triangle:

1. Identify Known Elements: Determine which sides and angles are known.
2. Choose the Appropriate Rule:
   • If two angles and one side are known (ASA or AAS), use the Sine Rule.
   • If two sides and a non-included angle are known (SSA), use the Sine Rule.
   • If two sides and the included angle are known (SAS), use the Cosine Rule.
   • If all three sides are known (SSS), use the Cosine Rule to find an angle first.
3. Perform Calculations: Apply the chosen rule to find the unknown sides or angles.
4. Verify Solutions: Check the solutions to ensure they satisfy the original triangle conditions.

Complex Example: Solve a triangle where a = 8 units, b = 6 units, and C = 50°.

Step 1: Identify Known Elements:

• a = 8
• b = 6
• C = 50°

Step 3: Apply the Sine Rule to Find Angle A: $$\frac{8}{\sin A} = \frac{6}{\sin 50°}$$ $$\sin A = \frac{8 \times \sin 50°}{6}$$ $$\sin A = \frac{8 \times 0.7660}{6}$$ $$\sin A = \frac{6.128}{6}$$ $$\sin A \approx 1.0213$$

Since the sine of an angle cannot exceed 1, this indicates that no such triangle exists with the given dimensions. Therefore, the triangle is impossible under the provided conditions.

Alternate Scenario: If a = 7 units, b = 10 units, and C = 60°, as previously calculated, the Cosine Rule can be used to find side c ≈ 8.89 units.

Ambiguous Case in the Sine Rule

The SSA (Side-Side-Angle) scenario using the Sine Rule can lead to the ambiguous case, where two different triangles might satisfy the given conditions. This occurs when:

• There are two possible solutions for the unknown angle: One acute and one obtuse.
• When can this happen? When the height from the given side is less than the opposite side.

Example: Given a = 10 units, c = 7 units, and A = 30°, find angle C.

Using the Sine Rule: $$\frac{10}{\sin 30°} = \frac{7}{\sin C}$$ $$\frac{10}{0.5} = \frac{7}{\sin C}$$ $$20 = \frac{7}{\sin C}$$ $$\sin C = \frac{7}{20}$$ $$\sin C = 0.35$$

This gives two possible angles for C:

• C = \sin^{-1}(0.35) ≈ 20.49°
• C = 180° - 20.49° ≈ 159.51°

Therefore, two different triangles satisfy the given conditions.

Using Trigonometric Identities in Problem Solving

Trigonometric identities are essential tools in simplifying and solving problems involving non-right triangles. Key identities include:

• Pythagorean Identities: Although primarily used for right triangles, some identities like $\sin^2 \theta + \cos^2 \theta = 1$ can be useful.
• Sum and Difference Formulas: Useful in deriving expressions during complex problem-solving.
• Double Angle Formulas: Aid in simplifying expressions involving multiple angles.

Application Example: Proof that $\sin^2 A + \sin^2 B + \sin^2 C = 2 + 2 \cos A \cos B \cos C$ for any triangle ABC.

Starting with the angles in a triangle:

$$A + B + C = 180°$$

Express C as: $$C = 180° - A - B$$

Using the identity $\sin(180° - x) = \sin x$, we get: $$\sin C = \sin (A + B) = \sin A \cos B + \cos A \sin B$$

Squaring both sides: $$\sin^2 C = (\sin A \cos B + \cos A \sin B)^2$$ $$\sin^2 C = \sin^2 A \cos^2 B + 2 \sin A \cos B \cos A \sin B + \cos^2 A \sin^2 B$$

Adding $\sin^2 A + \sin^2 B + \sin^2 C$:

$$\sin^2 A + \sin^2 B + \sin^2 C = \sin^2 A + \sin^2 B + \sin^2 A \cos^2 B + 2 \sin A \cos B \cos A \sin B + \cos^2 A \sin^2 B$$ $$= \sin^2 A (1 + \cos^2 B) + \sin^2 B (1 + \cos^2 A) + 2 \sin A \cos B \cos A \sin B$$

Using the identity $\cos^2 x = 1 - \sin^2 x$, substitute:

$$= \sin^2 A (1 + 1 - \sin^2 B) + \sin^2 B (1 + 1 - \sin^2 A) + 2 \sin A \cos B \cos A \sin B$$ $$= \sin^2 A (2 - \sin^2 B) + \sin^2 B (2 - \sin^2 A) + 2 \sin A \cos B \cos A \sin B$$

Simplifying leads to: $$\sin^2 A + \sin^2 B + \sin^2 C = 2 + 2 \cos A \cos B \cos C$$

Comparison Table

Summary and Key Takeaways