📐 Unlocking Any Triangle with Sine & Cosine Rules!

Ever wondered how pilots calculate distances between waypoints or how engineers determine the exact length of support beams in slanted roofs? **Non-right-angled trigonometry** gives us those answers! By mastering the Sine and Cosine Rules, you can solve *any* triangle that is not restricted to 90° angles—an essential skill for O Level exams and real-world applications.

Main Concept: Sine & Cosine Rules

The **Sine Rule** links the ratios of sides and opposite angles in any triangle:

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

It is most useful when you know $\triangle$ — two angles & one side (AAS/ASA) or two sides & a non-included angle (SSA). The **Cosine Rule** extends Pythagoras to non-right triangles:

a^2 = b^2 + c^2 - 2bc\cos A.

Use it when you have three sides (SSS) or two sides with the included angle (SAS). **Remember:** choose the rule that fits the given information to reduce mistakes and save exam time!

If two angles are known, find the third quickly using $A+B+C=180^{\circ}$ , then apply the Sine Rule.
Use the Cosine Rule first in SAS cases; once a side is found, switch to the Sine Rule for remaining angles.
Always keep your calculator in degree mode for O Level papers!

Worked Example: Finding a Side with the Cosine Rule

Problem: In $\triangle ABC$, sides

b=7\,\text{cm}

c=9\,\text{cm}

, and included angle

A=60^{\circ}

. Find the length of side

a

opposite

A

Step 1: Apply the Cosine Rule
$a^2 = b^2 + c^2 - 2bc\cos A$

$a^2 = 7^2 + 9^2 - 2(7)(9)\cos 60^{\circ}.$
Step 2: Compute
$a^2 = 49 + 81 - 126 \times 0.5 = 130 - 63 = 67.$
Step 3: Take the square root
$a = \sqrt{67} \approx 8.19\,\text{cm}.$

Non-right-angled trigonometry is a powerful toolkit—practice a variety of problems, and soon choosing between the Sine and Cosine Rules will feel **automatic**. Keep solving, stay curious, and ace that exam!