One triangle. Three rules.

SOHCAHTOA only works with a right angle. Most triangles haven't got one. So take an ordinary triangle — sides $8$ and $5$ with $60^\circ$ between them — and watch where the general rules come from.

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Capitals for angles, lower case for sides — and each side takes the name of the angle opposite it. Side $a$ sits across from angle $A$. Every rule in this chapter pairs them exactly that way, so get the labels right and the formulas follow.

One altitude, measured two ways, and the rule falls out. Use it when you know a side and the angle opposite it — each side paired with its own angle.

At $A = 90^\circ$, $\cos A = 0$ and this is Pythagoras. The $-2bc\cos A$ is the correction for every other angle: it shrinks $a$ when $A$ is acute and stretches it when $A$ is obtuse. Back at $60^\circ$, $a^2 = 64 + 25 - 40 = 49$, so $a = 7$.

Two sides and the angle between them. The altitude is back: its height is $8\sin 60^\circ$, so the area is $\tfrac{1}{2} \times 5 \times 8\sin 60^\circ = 10\sqrt{3} \approx 17.3$. Same idea as half base × height — the sine just finds the height for you.

Given $A = 40^\circ$, $b = 12$ and $a = 9$: swing a side of length 9 from $C$ and it meets the base line twice. $\sin B = 0.857$ gives $B = 59.0^\circ$ — but $\sin(180^\circ - B)$ is the same, so $B = 121.0^\circ$ fits too. Always check for the second triangle.

Sides meet their opposite angles.

Finding angles as well as sides, bearings problems, and twelve graded exam questions with full worked solutions are waiting.

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