Digital SAT Math · Geometry & Trigonometry

Right triangle trigonometry

Every trig item on this skill lives inside one right triangle. Name the reference angle, label opposite / adjacent / hypotenuse relative to that angle only, then pick the single ratio that holds the side you know and the side (or value) you want. The test’s favorite freebie is the cofunction identity $\sin(x^\circ) = \cos(90^\circ - x^\circ)$ — when two acute angles sum to 90°, you often copy a number instead of computing one.

On the test

Recognition cues: right triangle; opposite / adjacent / hypotenuse; $\sin$, $\cos$, $\tan$; angle of elevation; 30-60-90 or 45-45-90; two angles that sum to $90^\circ$.

Where this skill ends and its neighbours begin

Side lengths without a trig ratio (hidden right triangles, triples, 3D slant height) live on Pythagorean theorem & applications. Similarity and corresponding sides live on triangles. Radians, the unit circle, and signs by quadrant live on trigonometry beyond basic right triangles — anything solvable with SOH-CAH-TOA on one right triangle stays here. Angle relationships that never form a trig ratio (parallel lines, exterior angles) live on lines & angles.

Pattern recognition

Sort by what the stem hands you:

1. Ratio from sides — three sides (or two plus a right angle) are known; read $\sin$, $\cos$, or $\tan$ of a named acute angle.
2. Side from ratio — an angle and one side are known; solve for another side with the matching ratio (often a calculator item).
3. Complementary cofunction — two angles sum to $90^\circ$, or two acute angles of one right triangle; $\sin$ of one equals $\cos$ of the other.
4. Special right triangle — angles are $30^\circ$-$60^\circ$-$90^\circ$ or $45^\circ$-$45^\circ$-$90^\circ$; exact side ratios replace the calculator.

Method

1. Mark the reference angle. Circle the acute angle the stem names (or the elevation angle). Everything else is relative to that mark.
2. Label opposite, adjacent, hypotenuse. Hypotenuse = side across from $90^\circ$. Opposite = side across from the reference angle. Adjacent = the remaining leg (touches the reference angle, not the hypotenuse).
3. Pick the ratio that contains the known and the wanted.
   • $\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$
   • $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$
   • $\tan\theta = \dfrac{\text{opp}}{\text{adj}}$
4. Solve. Cross-multiply, or multiply both sides by the denominator. For cofunction items, convert rather than compute: if $x + y = 90$, then $\sin x^\circ = \cos y^\circ$.
5. Special-triangle shortcut when the angles are $30/45/60$. Use the fixed side ratios below instead of a decimal approximation unless the stem asks for a rounded length from a non-special angle.

Practice

Answer before opening the explanation. Five items depend on a labeled right triangle. Two items are student-produced response. Every wrong choice below is a specific error with a name — wrong reference angle, hyp used as a leg, complement applied to tangent, sine/cosine/tangent swapped, or calculator mode.

Common mistakes

1. Opposite/adjacent labeled for the wrong angle — sides marked once for $A$, then reused for $B$. Relabel every time the reference angle changes.
2. Hypotenuse used as a leg — $\tan\theta = \text{opp}/\text{hyp}$ or a side solved as though the given length were adjacent when it is the hypotenuse.
3. Complement identity applied to tangent — assuming $\tan(x^\circ) = \tan(90^\circ - x^\circ)$. The truth is $\tan(90^\circ - x^\circ) = \cot(x^\circ) = 1/\tan(x^\circ)$.
4. Calculator in radians — $\sin 42$ evaluated as $\sin(42\text{ rad})$ instead of $42^\circ$. Force degree mode (or a degree symbol in Desmos) before every evaluation.
5. Sine/cosine/tangent swapped — correct sides, wrong letter of SOH-CAH-TOA (e.g. reporting adjacent/hyp when the stem asked for sine).
6. Special-triangle radicals mixed — $\sqrt{2}$ used on a $30$-$60$-$90$ long leg, or $\sqrt{3}$ on a $45$-$45$-$90$ hypotenuse.
7. Stopping at the unit ratio — reporting $5/13$ when the hypotenuse is $26$ and the question asked for a side length of $10$.
8. Inverting the ratio — solving $\text{opp} = \text{hyp} / \sin\theta$ when the setup was $\text{opp} = \text{hyp}\cdot\sin\theta$.
9. Rounding too early — truncating $\tan 32^\circ$ before multiplying; round only the final length when the stem says “nearest tenth.”

FAQ

Do I need to memorize exact values of $\sin 30^\circ$, $\cos 45^\circ$, …? Yes for the special angles that appear as side-ratio items: $\sin 30^\circ = 1/2$, $\cos 30^\circ = \sqrt{3}/2$, $\sin 60^\circ = \sqrt{3}/2$, $\cos 60^\circ = 1/2$, $\sin 45^\circ = \cos 45^\circ = \sqrt{2}/2$. The side patterns $1:1:\sqrt{2}$ and $1:\sqrt{3}:2$ give the same values and are often faster.

When do I use Pythagoras instead of trig? When you already know two sides and need the third, and no angle (other than the right angle) is in play — that is the Pythagorean-theorem skill. On this page Pythagoras appears only as a bridge (e.g. turn $\tan A = 5/12$ into a $5$-$12$-$13$ triangle so you can read $\sin A$).

Does $\sin(x^\circ) = \cos(90^\circ - x^\circ)$ work in radians too? Yes as an identity: $\sin x = \cos(\pi/2 - x)$. On the Digital SAT, radian forms of that identity belong with unit-circle work on trigonometry-extended. Every item on this page states angles in degrees.

How do I enter a radical on a grid-in? Most SPR items on this skill ask for a decimal (nearest tenth) or a plain integer short leg. If a stem ever demands an exact radical, the Bluebook accepts equivalent forms — but our practice SPRs here are integers or one-decimal answers.

Angle of depression vs elevation? Both are measured from the horizontal. Elevation looks up; depression looks down. The right triangle you draw is the same shape — the horizontal run is adjacent and the vertical rise/drop is opposite when the angle sits at the observer.

Editorial team · last reviewed 2026-08-12