In A-Level Pure Mathematics, a deep understanding of how radians, arc length, and sector area apply to triangles is essential. This comprehensive guide focuses on these applications, particularly in calculating lengths and angles in triangles using the sine and cosine rules. We will also explore the calculation of areas of sectors and segments within triangles, complete with detailed exercises and solutions.

Understanding the Basics

• Radians in Triangles: Radians, a natural way of measuring angles, are particularly useful in the context of triangles. They provide a direct relationship between the angle and the arc length of a circle.
• Arc Length and Sector Area: These concepts are pivotal in understanding the geometric properties of triangles, especially when they intersect with or are part of circular sectors.

Utilising Sine and Cosine Rules

Sine Rule

The sine rule is a powerful tool in solving non-right-angled triangles. It states that the ratio of the length of a side to the sine of its opposite angle is constant across all sides and angles of the triangle.

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

This rule is particularly useful in scenarios where two angles and one side are known, or two sides and a non-included angle are known.

Cosine Rule

The cosine rule is applicable to any triangle, whether right-angled or not. It is used to find a side length when two sides and the included angle are known, or an angle when all sides are known.

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

This rule is essential when dealing with scenarios where the sine rule cannot be applied directly, such as finding the third side of a triangle when two sides and the included angle are known.

Calculating Sector Areas and Arc Lengths in Triangles

Sector Area

The area of a sector formed by a triangle can be calculated using the formula:

where $\theta$ is the central angle in radians.

Courtesy image of Math Central

Arc Length

The length of the arc can be determined by:

This formula is integral in understanding the perimeters involving circular arcs in triangles.

Image courtesy of thirdspacelearning

Practical Examples

Example 1: Calculating Triangle Angles Using Radians

Given a triangle with sides of lengths 7 cm, 5 cm, and 3 cm, calculate the angle opposite the longest side using the cosine rule.

Solution:

• Given Sides: $a = 5$ cm, $b = 3$ cm, $c = 7$ cm (longest side).
• Cosine Rule:

• Angle $C$ in Radians:

Using the inverse cosine of $-0.5$, the angle opposite the longest side is calculated as $\cos^{-1}(-0.5) = \frac{2\pi}{3}$ radians.

• Result:

The angle opposite the longest side $c = 7$ cm is $\frac{2\pi}{3}$ radians, which corresponds to $120^\circ$.

Example 2: Finding the Area of a Sector within a Triangle

A triangle has two sides of length 6 cm and 8 cm, with an included angle of 60 degrees. Find the area of the sector formed by these two sides.

Solution:

• Angle Conversion: $60^\circ$ to radians is $\frac{\pi}{3}$.
• Sector Area Formula:

$A = \frac{1}{2} r^2 \theta$

• Given:

Radius $(r)$ = 6 cm,

Central angle $(\theta) = \frac{\pi}{3}$ radians.

• Calculation:

$A = \frac{1}{2} \times 6^2 \times \frac{\pi}{3}$

$A = 6\pi \, \text{cm}^2$

Result: The area of the sector formed by the two sides of the triangle is $6\pi$square centimeters, showcasing how to apply sector area formula in a triangle-related context.