Sine rule Notes

Using A Consistent Triangle Labeling Makes Formulas Work

1. When working with non-right-angled triangles, a standard labeling convention helps everyone communicate clearly and use the same formulas correctly.
2. Label the vertices with capital letters $A$, $B$, $C$.
3. The interior angle at each vertex uses the same letter ($\angle A$, $\angle B$, $\angle C$).
4. The side opposite each angle uses the same letter in lowercase ($a$, $b$, $c$).

The Sine Rule Links Sides To Their Opposite Angles

1. The sine rule (also called the sine law) is a relationship that is true in every triangle (not just right-angled ones).
2. It says that the ratio "side divided by sine of its opposite angle" is constant within a triangle.

Why The Sine Rule Is True (Idea Using A Perpendicular Height)

A useful way to understand the sine rule is to drop a perpendicular height in the triangle and connect it to right-triangle trigonometry.

1. Let $h$ be the height from $A$ to $BC$.
   1. In right triangle $ABD$, $\sin B = \dfrac{h}{c}$, so $h=c\sin B$.
   2. In right triangle $ACD$, $\sin C = \dfrac{h}{b}$, so $h=b\sin C$.
2. Since both expressions equal the same height $h$: $$c\sin B=b\sin C$$
3. Divide both sides by $bc$: $$\frac{\sin B}{b}=\frac{\sin C}{c}$$
4. Repeating this idea with a different altitude gives the full statement $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$

When To Use The Sine Rule (And When Not To)

1. The sine rule is most useful when the information you have includes an angle and its opposite side.
2. In many problems, it is described as involving two sides and two angles.

Typical "Sine Rule Friendly" Data

You can usually use the sine rule immediately in these situations:

1. ASA or AAS: two angles and one side
2. SSA: two sides and an angle opposite one of them (but see the warning about ambiguity)

When Another Rule Is Better

If you do not have an opposite pair, the sine rule cannot start the problem directly. In that case the cosine rule is often the correct tool, especially for:

1. SSS (three sides)
2. SAS (two sides and the included angle)

How To Solve Problems With The Sine Rule (A Reliable Procedure)

1. Label the triangle using $A,B,C$ and $a,b,c$ so "opposite" is clear.
2. Choose two matching fractions from the sine rule, involving your unknown.
3. Substitute values and solve.
4. If finding an angle, use inverse sine ($\arcsin$), then check whether a second angle is possible (ambiguous case).
5. If needed, use the angle sum of a triangle: $A+B+C=180^\circ$.

The Ambiguous Case (SSA) And How To Check It

1. The ambiguous case happens when you know two sides and an angle that is not the included angle (SSA).
2. Because $\sin \theta = \sin(180^\circ-\theta)$, the equation $$\sin C = k$$ can correspond to two angles in $(0^\circ,180^\circ)$:
   1. $C_1=\arcsin(k)$
   2. $C_2=180^\circ-\arcsin(k)$
3. To decide whether there are 0, 1, or 2 solutions, check whether each candidate angle makes sense with the other information:
   1. Angles must be positive.
   2. The total must satisfy $A+B+C=180^\circ$.
   3. Sides must remain consistent (largest side opposite largest angle).

Bearings Create Triangles Where The Sine Rule And Cosine Rule Are Powerful

A bearing is a direction measured clockwise from North, written as a three-digit angle (for example, $060^\circ$).

1. If two boats leave the same point on different bearings, their paths form two sides of a triangle.
2. The angle between their courses is the difference of the bearings (when they start at the same point).
3. If one boat travels on a bearing of $060^\circ$ and the other on $135^\circ$, so the angle between paths is $$135^\circ-60^\circ=75^\circ$$
4. Once you have a triangle with two sides and the included angle (SAS), you can find the third side with the cosine rule, then use the sine rule to find remaining angles if needed.