Can You Use the Sine and Cosine Rules in a Right-Angled Triangle?

A right-angled triangle or a triangle with one right angle (90 degrees) plays a fundamental role in trigonometry. The definitions of sine and cosine for any angle within this triangle are derived based on the ratios of the triangle's sides. Let's explore the sine and cosine rules and their application in right-angled triangles.

Defining Sine and Cosine in a Right-Angled Triangle

Consider a non-right angle, symbolized by θ, in a right-angled triangle. By definition:

Sine of θ sinθ Opposite Side / Hypotenuse Cosine of θ cosθ Adjacent Side / Hypotenuse

Using the Cosine Rule

The Cosine Rule is applicable to any triangle, but it also works for right-angled triangles. Let's use the Cosine Rule to find the length of the hypotenuse of a right-angled triangle where both the adjacent and opposite sides are exactly one unit long.

Consider a right-angled triangle with sides a, b, and c. Here, a is the hypotenuse:

a √(b2 c2 - 2bc cosα)

Let's apply this to a right-angled triangle where b 1 and c 1, and the angle between b and c is 90 degrees:

α 90°

Step-by-step calculation:

Calculate the cosine of 90°: cos90° 0

Substitute the values into the Cosine Rule formula:

a √(12 12 - 2(1)(1)cos90°)

Simplify the expression:

a √(1 1 - 2(0))

a √(2)

This result aligns with the known diagonal of a unit square, which is √2.

Understanding Sine and Cosine in Right-Angled Triangles

In a right-angled triangle, the sides are named as follows:

Base Altitude (Perpendicular) Hypotenuse

For a non-right angle θ:

sinθ Opposite Side / Hypotenuse cosθ Adjacent Side / Hypotenuse

The Power of the Sine and Cosine Rules

The Sine and Cosine rules extend beyond right-angled triangles and apply to any type of triangle. Here are the formulas:

LAW OF COSINES

The Law of Cosines states:

c2 a2 b2 - 2ab cosC

Here, C is the angle between sides a and b. If c is the hypotenuse, then C 90° and cosC 0. Therefore, the equation simplifies to:

c2 a2 b2

which is the Pythagorean Theorem.

LAW OF SINES

The Law of Sines states:

a/sinA b/sinB c/sinC

If c is the hypotenuse, then C 90° and sinC 1. Thus, the equation becomes:

a/sinA b/sinB c

Since c/sinC 1, it means: