Foundations — Law of sines — proof and applications (ambiguous case)

This is the "load your toolbox" page. The parent note uses a pile of symbols and facts without stopping to build them. Here we stop and build each one, in an order where every new idea leans only on the ones before it.

0. What a triangle's letters even mean

Before any formula, we must agree on the naming habit the whole topic silently assumes.

Look at the figure. The amber side $a$ is the one your eye jumps to when you stand at corner $A$ and look straight across the triangle — it's the "far wall". The two sides that do touch corner $A$ (namely $b$ and $c$) are its "arms". This "far wall = opposite side" picture is the entire reason the law pairs $a$ with $A$: it always couples a side with the corner facing it.

1. What an angle is, and degrees

We measure that swing in degrees, written with a little circle ${}^{\circ }$.

• A full spin all the way around = $360^{\circ }$.
• A quarter turn (a perfect square corner, the "L" shape) = $90^{\circ }$. This special angle is called a right angle, and we mark it with a little square in the corner.
• A dead-straight line = half a spin = $180^{\circ }$.

That last fact hides the single most-used rule about triangles:

Why the topic needs this: in the ambiguous case we get a candidate angle and immediately test whether the three angles still fit under $180^{\circ }$. Without this rule there would be no way to reject an impossible triangle.

2. The right triangle and the word sine

Everything in the proof rests on one trig fact, so we build it slowly.

Now the star of the show:

Two edge readings you must be comfortable with:

• $\sin 0^{\circ }=0$ (a flattened triangle has zero height, so the opposite side has length $0$).
• $\sin 90^{\circ }=1$ (the opposite side is the hypotenuse, ratio $=1$).
• So $\sin \theta$ always lands between $0$ and $1$ for an angle in a triangle. Remember this ceiling of $1$ — a "sine bigger than $1$" is the alarm bell for "no triangle exists" in the parent's Example 3.

Why the topic needs sine: the whole proof drops a height $h$, and in each right triangle it reads $\sin (\text{angle})=h÷(\text{a side})$. Sine is the only trig tool the derivation uses — which is why the parent stresses "that's the only trig fact we need."

3. ${\sin }^{-1}$ — undoing the sine (arcsine)

The proof gives you a number for $\sin B$ and you must recover the angle $B$. That reverse move needs its own symbol.

Here is the subtlety that seeds the entire ambiguous case:

4. Supplementary angles — why the same sine has TWO owners

Look at the figure: build a right triangle giving height $h$ from an acute angle $\theta$, then swing to the obtuse angle $180^{\circ }-\theta$. The height climbed is identical — same opposite side, same hypotenuse. Hence:

So the equation $\sin B=0.8333$ has two owners inside a triangle: an acute $B_1={\sin }^{-1}(0.8333)\approx 56.44^{\circ }$ and an obtuse $B_2=180^{\circ }-56.44^{\circ }\approx 123.56^{\circ }$.

Why the topic needs this: this single identity is the root of the SSA "ambiguous case" — it's why the swinging-hinge door can hit the wall in two places. It's also the rescue for obtuse-angle triangles in the proof: the altitude relation $h=b\sin A$ survives because $\sin (180^{\circ }-A)=\sin A$. See Sine of Supplementary Angles.

5. The circumcircle and $R$

The right-hand side of the law, $2R$, is pure geometry — so we build the circle.

Why the topic needs this: the parent's headline result is that the common ratio $\frac{a}{\sin A}$ equals $2R$. So $R$ isn't decoration — it's the number the whole law computes. See Circumcircle and Circumradius R.

Two circle facts the proof leans on (built fully in Inscribed Angle Theorem):

• Angle in a semicircle is $90^{\circ }$: if a triangle's longest side is a diameter of its circumcircle, the opposite corner is a right angle. (This is what makes triangle $BCD$ in the proof a right triangle.)
• Inscribed-angle / same-chord equality: two corners on a circle that look at the same chord see it at the same angle. (This is why $\angle D=\angle A$ in the proof.)

6. Reading fractions, $\approx$, and $⟹$

Tiny notation the parent sprinkles everywhere:

• $\frac{a}{\sin A}$ means "the length $a$ divided by the number $\sin A$".
• $\approx$ means "approximately equal" — we rounded (e.g. $\sin 40^{\circ }\approx 0.6428$).
• $⟹$ means "therefore / which forces". "$P⟹Q$" reads "$P$, so $Q$."
• $\ge$ means "greater than or equal to"; $<$ means "strictly less than".

How these foundations feed the topic

Equipment checklist

Test yourself — cover the right side and answer aloud before revealing.

Connections

• Parent: Law of Sines — everything here loads its toolbox.
• Sine of Supplementary Angles — the $\sin (180^{\circ }-\theta )=\sin \theta$ engine.
• Circumcircle and Circumradius R — meaning of $R$ and $2R$.
• Inscribed Angle Theorem — powers the $2R$ step.
• Law of Cosines — the sibling law for SAS/SSS.
• Solving Oblique Triangles — where these tools get chosen.
• Area of a Triangle — another home for $R$.