Law of Sines - garden

When There Is No Right Angle

SOH-CAH-TOA has one hard requirement: a $90°$ angle. But most triangles do not have one.

The law of sines removes that limit. It solves any triangle, right-angled or not.

The Law

First, the naming. In any triangle, each side is labelled with the lowercase version of the angle across from it:

side $a$ sits opposite angle $A$
side $b$ sits opposite angle $B$
side $c$ sits opposite angle $C$

Now watch the three ratios on the right as the triangle changes shape. Each side, divided by the sine of its opposite angle, gives the same number every time.

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

The shape can be anything at all, and the three ratios still lock together.

Why “Opposite” Matters

The pairing is the heart of it: each side is tied to the angle facing it.

The bigger the angle, the longer the side across from it. The law makes that relationship exact.

This is why you can never use just any side with just any angle. You always work with a matched pair: a side and the angle opposite it.

When You Can Use It

The law needs one complete pair ( $\frac{\text{side}}{\sin\text{angle}}$ ) to get started. That happens in two situations:

You know	Name
two angles and any side	AAS / ASA
two sides and an angle opposite one of them	SSA

Example: Two Angles and a Side

In a triangle, $A = 40°$ , $B = 75°$ , and side $a = 10$ . Find side $b$ .

Side $a$ and angle $A$ form a complete pair, so set $b$ against it:

\begin{aligned} \frac{b}{\sin 75°} &= \frac{10}{\sin 40°} \\[0.5em] b &= \frac{10\sin 75°}{\sin 40°} \\[0.5em] &\approx 15.0 \end{aligned}

Finding an Angle

Rearranged the other way, the law finds a missing angle.

Given $a = 12$ , $b = 9$ , and $A = 55°$ , find $B$ .

\begin{aligned} \frac{\sin B}{9} &= \frac{\sin 55°}{12} \\[0.5em] \sin B &= \frac{9\sin 55°}{12} \approx 0.614 \\[0.5em] B &= \arcsin(0.614) \approx 37.9° \end{aligned}

Careful: when you solve for an angle with $\arcsin$ , a second angle (its supplement) can sometimes also fit. This is the ambiguous case, and it means an SSA setup may describe two different triangles.

The Deeper Reason

That shared value is not random. It equals the diameter of the circle that passes through all three corners of the triangle (the circumscribed circle of radius $R$ ):

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

Every triangle sits inside exactly one such circle, and that is what all three ratios are quietly measuring.