Right triangles and trigonometry is one of the four Geometry and Trigonometry subtopics, and it rewards students who know a small set of ratios cold. This guide walks through SOH-CAH-TOA, the two special right triangles worth memorizing, when the Pythagorean theorem beats a trig ratio, and two complete worked examples covering both approaches.
SOH-CAH-TOA: the three ratios you need
Every right-triangle trig question comes back to three ratios, defined relative to one of the non-right angles:
- \(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\)
- \(\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)
- \(\tan\theta = \frac{\text{opposite}}{\text{adjacent}}\)
The hardest part isn't the formulas — it's correctly labeling which leg is "opposite" and which is "adjacent" relative to the angle you're using. Redraw the triangle, mark the angle you care about, and label the two legs relative to that specific angle before you touch a formula. This matters especially when a figure is rotated on the page, since the "bottom" leg as drawn isn't necessarily the adjacent leg for the angle you're solving for.
Special right triangles you should have memorized
Two triangle shapes appear constantly enough that memorizing their side ratios saves real time over building them from trig ratios each time.
| Triangle | Angle measures | Side ratio |
|---|---|---|
| 45-45-90 | \(45^\circ, 45^\circ, 90^\circ\) | \(x, x, x\sqrt{2}\) |
| 30-60-90 | \(30^\circ, 60^\circ, 90^\circ\) | \(x, x\sqrt{3}, 2x\) |
These ratios are actually printed on the digital SAT's reference sheet, which makes them easy to look up mid-test — but recognizing when a triangle is one of these two shapes in the first place is the real skill, and that recognition only comes from practice. Our guide to the geometry formulas the SAT does not give you explains exactly which related formulas you still have to know from memory.
The Pythagorean theorem: when to use it instead
When you know two sides of a right triangle but no angle, trig ratios aren't the tool — the Pythagorean theorem is: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse. Use trig when you have an angle and need a side; use the Pythagorean theorem when you have two sides and need the third. Mixing these up is one of the most common time-wasters on this subtopic — students sometimes try to force a sine or cosine ratio onto a problem that never gave them an angle at all, when a two-line Pythagorean calculation would have finished the job.
Worked example: finding a missing side with trig
A right triangle has a hypotenuse of 20 units. One of the acute angles measures \(32^\circ\). Find the length of the side opposite that angle, rounded to the nearest tenth.
- Identify what's given and what's asked: you have the hypotenuse and an angle, and you want the side opposite that angle. That's a sine relationship.
- Write the ratio: \(\sin(32^\circ) = \frac{\text{opposite}}{20}\).
- Solve for the opposite side: \(\text{opposite} = 20 \times \sin(32^\circ)\).
- Using a calculator, \(\sin(32^\circ) \approx 0.530\), so \(\text{opposite} \approx 20 \times 0.530 = 10.6\) units.
The built-in Desmos calculator available throughout the SAT Math section handles the decimal trig value instantly, so the real work is setting up the right ratio in step two — that's where points are actually won or lost.
Worked example: finding a missing side with the Pythagorean theorem
A right triangle has legs of length 9 and 12. Find the length of the hypotenuse.
- Identify what's given and what's asked: two sides (both legs) are known, and no angle is given or needed. That rules out trig ratios and points straight to the Pythagorean theorem.
- Write the formula: \(a^2 + b^2 = c^2\), with \(a = 9\) and \(b = 12\).
- Substitute and simplify: \(9^2 + 12^2 = 81 + 144 = 225\).
- Take the square root: \(c = \sqrt{225} = 15\).
Notice this is a scaled-up 3-4-5 right triangle — 9 and 12 are each three times 3 and 4, so the hypotenuse is three times 5. Recognizing common Pythagorean triples like 3-4-5, 5-12-13, and 8-15-17 (and their multiples) lets you skip the arithmetic entirely on some questions.
Common traps on right triangle questions
- Confusing opposite and adjacent when the given angle isn't at the "bottom" of the triangle as drawn.
- Using the Pythagorean theorem when an angle is actually given and a trig ratio would be faster.
- Forgetting that \(\sin\) and \(\cos\) of complementary angles are related: \(\sin\theta = \cos(90^\circ - \theta)\), which sometimes shortcuts a problem entirely.
- Rounding too early in multi-step problems, which compounds error by the final answer.
- Missing a Pythagorean triple in disguise and grinding through unnecessary decimal arithmetic instead of scaling a familiar ratio.
Quick tip: if a triangle's angles are 30-60-90 or 45-45-90, skip trig ratios entirely and use the side-ratio shortcuts — they're faster and less error-prone.
Frequently asked questions
Does the SAT test trigonometry beyond right triangles?
Yes — some questions extend these ratios using the unit circle, which is really just SOH-CAH-TOA applied to angles beyond 90 degrees.
Do I need a scientific calculator for trig questions?
No separate calculator is required — the embedded Desmos calculator available throughout the Math section computes sine, cosine, and tangent values directly.
How do I decide between trig and the Pythagorean theorem quickly?
Check what you're given before you write anything. Two sides and no angle means Pythagorean theorem; one side and one angle means a trig ratio. That single check, done first, prevents most of the wasted time on this subtopic.
How do I get faster at labeling triangle sides correctly?
Timed repetition is what builds that instinct. Our SAT practice questions for Right Triangles and Trigonometry include full worked explanations so you can see exactly how each side should have been labeled.