Relative to a chosen angle $\theta$ (not the right angle):

• Hypotenuse — the longest side, always opposite the right angle.
• Opposite — the side directly across from $\theta$.
• Adjacent — the remaining side, next to $\theta$.

Relabel the triangle every time you switch which angle you're working with — "opposite" and "adjacent" have no fixed meaning until an angle is chosen.

Identify which two sides are involved (one known, one unknown), then pick whichever ratio contains exactly those two. If the hypotenuse isn't involved at all, it must be tan.

Set up the ratio, then rearrange so the unknown side is alone. For example, if $\angle A = 34°$, hypotenuse $AC = 12$ cm, and $BC$ (opposite $A$) is unknown:

If the unknown side is on the bottom of the ratio instead (e.g. it's the hypotenuse), cross-multiply before dividing rather than trying to divide directly by a trig value in your head.

Rearrange to isolate the ratio itself, then undo it with the inverse function ($\sin^{-1}$, $\cos^{-1}$ or $\tan^{-1}$). For a ramp rising $1.4$ m over a horizontal run of $8.6$ m:

Both are measured from a horizontal line, never a vertical one:

• Elevation — looking up from the observer to a point above.
• Depression — looking down from the observer to a point below.

Because the two horizontal sightlines are parallel, alternate angles mean the angle of elevation from the ground equals the angle of depression from the point above — so a diagram drawn from either viewpoint gives the same working triangle.

Some questions don't give enough information to solve one triangle directly — a length found in a first triangle has to be carried into a second. Work through them in the order the information allows, and keep the unrounded value from the first triangle for use in the second.

Example: From point $P$ at ground level, the angle of elevation to the top of a tower is $22°$. Walking $15$ m directly towards the tower to point $Q$, the angle of elevation increases to $35°$. Find the height $h$ of the tower.

Let $x$ be the distance from $Q$ to the base of the tower. Both right-angled triangles share the same height $h$:

Rearranging both for $h$ and setting them equal: $x\tan(35°) = (x+15)\tan(22°)$, so

This "two triangles, one shared side" structure is one of the most common Extended-tier trigonometry setups — draw both triangles on the same diagram before writing any equation.

A cliff $AB$ is $45$ m high. From a boat $C$ on the water, the angle of elevation to the top $A$ is $28°$. Find the distance $BC$ from the boat to the base of the cliff.

Solution

Relative to the $28°$ angle at $C$: the height ($45$ m) is Opposite, $BC$ is Adjacent — use tangent.