Sine Rule

Introduction to Sine and Cosine Rules

In trigonometry, the Sine and Cosine Rules are incredibly powerful tools that extend the basic trigonometric ratios (Sin, Cos, Tan) to any triangle—not just right-angled triangles. This makes them extremely versatile and useful for solving a wide variety of problems.

The Sine Rule

The Sine Rule is used when you know:

• Two angles and one side ($AAS$ or $ASA$).
• Two sides and a non-included angle ($SSA$). The formula for the Sine Rule is:

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

Where:

• $a$, $b$, and $c$ are the lengths of the sides of the triangle.
• $A$, $B$, and $C$ are the angles opposite those sides.

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The Sine Rule: Finding an Unknown Side

What Information do you need to be given?

• Two angles and the length of a side.

What is the Formula?

The Sine Rule is given by:

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

Where $a, b$, and $c$ are the sides of the triangle, and $A, B$, and $C$ are the angles opposite these sides.

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The Sine Rule: Finding an Unknown Angle

What Information do you need to be given?

• Two lengths of sides and the angle not included (i.e., not between those two sides).

What is the Formula?

The Sine Rule can also be rearranged to find an unknown angle:

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

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