Sine rule

The Sine Rule

Introduction

• The Sine Rule is an important rule in trigonometry used for finding missing side lengths or angles in any triangle, not just right-angled triangles.
• It is expressed as a/sinA = b/sinB = c/sinC, where a, b and c are the sides of the triangle opposite to respective angles A, B and C.

Use Case 1: Finding an Unknown Side

• If you have a triangle with known angles and one known side, you can use the Sine Rule to find an unknown side.
• To find a side, the formula is rearranged as *a = bsinA / sinB*, where *b is the known side length, and A and B are the respective angles.

Use Case 2: Finding an Unknown Angle

• If you have a triangle with known side lengths and one known angle, you can use the Sine Rule to find an unknown angle.
• The rearranged formula for finding an angle is *A = arcsin[(asinB) / b]*, where *a and b are the known side lengths and B is the known angle.

Ambiguous Case of Sine Rule

• There’s an ambiguous case that may arise when applying the Sine Rule to find an unknown angle.
• This happens when the arcsin function in the calculator gives a principal value, which is acute (less than 90°), but there can also be a valid obtuse angle (between 90° and 180°) which also satisfies the Sine Rule.
• This ambiguity can be resolved with additional information or by using the rules of geometry.

Usefulness in Real Life

• The Sine Rule is not just confined to textbooks and exam questions but finds real-world application in fields such as engineering, physics, astronomy and computer graphics.

Tips for Remembering

• To avoid confusion while applying the Sine Rule, remember that the values being compared or being sought always sit opposite each other. Side a is opposite angle A, side b is opposite angle B, and so forth.