Triangles cheat sheet

Side/angle ordering and SOH CAH TOA, on one page

1 · Ordering sides & angles

Bigger side ↔ bigger angle across from it.

Across from = missing letter. Side \(YZ\) has no \(X\), so it's across from \(\angle X\).

  1. Match each side to its missing letter.
  2. Rank what you're given. Only two angles? Third \(=180°-\text{the other two}\).
  3. Carry the ranking across.

Example: \(XY=12,\ YZ=8,\ XZ=17\) → \(\angle X<\angle Z<\angle Y\).
\(\angle D=48°,\ \angle E=75°\) → \(\angle F=57°\) → \(EF

Trap: pairing an angle with a side that touches it.

2 · SOH CAH TOA

\[\sin=\frac{O}{H}\quad\cos=\frac{A}{H}\quad\tan=\frac{O}{A}\]

  1. Find the given angle (not the \(90°\)).
  2. Label: H across from the \(90°\); O across from the angle; A touches the angle.
  3. Know + want: O&H → sin · A&H → cos · O&A → tan.
  4. Unknown on top → multiply. On bottom → divide.
  5. Check: H is the longest side.

Ramp: know O = 3, want H → \(\sin5°=\frac{3}{L}\) → \(L=\frac{3}{\sin5°}\).
Ladder: know H = 20, want A → \(\cos70°=\frac{d}{20}\) → \(d=20\cos70°\).

Trap: \(3\sin5°\) is smaller than 3, so it can't be a hypotenuse.

Sources: Math Open Reference · Lippman & Rasmussen §5.5