Math · Geometry and Trigonometry

Right Triangles and Trigonometry on the Digital SAT

Right Triangles and Trigonometry questions combine two tools: the Pythagorean theorem for finding missing sides, and the trig ratios sine, cosine, and tangent for connecting sides to angles. Add the two special right triangles, 45-45-90 and 30-60-90, and you have essentially the whole skill. The Digital SAT reference sheet even includes both special triangle patterns, so you never have to reconstruct them from scratch.

These questions reward setup discipline. Sine, cosine, and tangent are defined relative to a specific angle, so "opposite" and "adjacent" change meaning depending on which acute angle you are working from. A large share of wrong answers on this skill are correct calculations of the wrong ratio: cos instead of sin, or the ratio for angle B when the question asked about angle A.

You should also expect Pythagorean triples like 3-4-5 and 5-12-13 to show up constantly, sometimes scaled by a factor. Recognizing 39 as 3 times 13 can turn a multi-step problem into arithmetic you can do in your head, and it is exactly the kind of shortcut the hardest questions are built around.

New to the Digital SAT? Start with how the test is structured and scored.

What the SAT actually tests

The Pythagorean theorem, a² + b² = c², including recognizing common triples like 3-4-5 and 5-12-13 and their multiples
The trig ratios via SOH-CAH-TOA: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent
Given one trig ratio of an acute angle, finding another ratio of the same angle or of the other acute angle
Special right triangles: 45-45-90 with sides x, x, x√2 and 30-60-90 with sides x, x√3, 2x
The complementary angle relationship sin(x°) = cos(90° − x°) in a right triangle
Trig ratios across similar right triangles: corresponding angles share the same sine, cosine, and tangent, so a ratio found in one triangle transfers to a similar one
Multi-step problems that chain a trig ratio with the Pythagorean theorem, such as finding a perimeter from one ratio and one side

Strategies that work

Draw and label the triangle yourself

Most trig questions on the Digital SAT are text-only. Sketch the right triangle, mark the right angle, and label which side is opposite and adjacent to the angle in the question. The labels depend on the angle, so redo them if the question switches from angle A to angle B.

Hunt for Pythagorean triples before computing

When you see side lengths like 13 and 5, or a hypotenuse of 39, check for a scaled 3-4-5 or 5-12-13 triangle before squaring anything. Spotting a triple saves time and avoids square-root arithmetic errors under pressure.

Turn a trig ratio into concrete sides

If sin A = 3/5, immediately sketch a right triangle with opposite 3, hypotenuse 5, and use the Pythagorean theorem to get the third side, 4. Now every other ratio for that triangle is just reading off two labeled sides, with no formulas left to confuse.

Recognize the two special triangles fast

A 45-45-90 triangle has sides x, x, x√2, and a 30-60-90 triangle has sides x, x√3, 2x, with the shortest side always opposite the smallest angle. Both are printed on the Digital SAT reference sheet, so you never have to memorize them. The win is recognizing the pattern quickly and spending your time on the setup instead of decoding the sheet.

Mistakes to avoid

Adding the squares when solving for a leg: the hypotenuse gets c², so a leg is √(c² − b²), not √(c² + b²)
Computing the trig ratio of the wrong acute angle, since opposite and adjacent swap between the two angles
Mixing up which ratio is which, most often answering with cos when the question asked for sin or tan
Placing √2 or √3 on the wrong side of a special right triangle, such as putting x√2 on a leg of a 45-45-90 triangle
Solving for an intermediate value like a scale factor and stopping there instead of answering the final question, such as a perimeter

Solve it with Desmos

Let the calculator do the work

The Digital SAT's built-in Desmos calculator can crack Right Triangles and Trigonometry questions in a few clicks. Here is the move that applies, step by step.

Solve Trig Equations in Degree ModeFlip Desmos to degrees, graph the trig equation, and click the crossing.

All 15 Desmos techniques

Practice with Nova, your Math coach

Answer right here and Nova walks you through it, just like in a drill. Difficulty labels are relative within this skill.

Tip: the Calculator button on each question opens the same Desmos graphing calculator you get on the real Digital SAT. Try solving these with it.

Question 1Easy

In right triangle ABC shown, the right angle is at C. The hypotenuse AB has length 13, and side BC has length 5.

What is the length of side AC?

Pick an answer to get instant coaching from Nova.

Question 2Medium

In right triangle ABC, the right angle is at C, and sin A = 3/5.

What is the value of tan A?

Pick an answer to get instant coaching from Nova.

Question 3Hard

In right triangle ABC, the right angle is at C, tan A = 5/12, and the hypotenuse AB has length 39.

What is the perimeter of triangle ABC?

Pick an answer to get instant coaching from Nova.

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The rest of Geometry and Trigonometry

Right Triangles and Trigonometry is one of 4 skills in Geometry and Trigonometry. The ideas build on each other, so it pays to learn them as a set rather than in isolation. See how Geometry and Trigonometry fits together.

Other Math skills to explore

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