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Geometry and Trig · Skill 3 of 4

SAT Right Triangles and Trigonometry Practice

Right triangle trigonometry on the digital SAT is three ratios, two special triangles, and one identity. The questions dress up differently, a ladder, a ramp, a coordinate grid, but every one resolves to picking the correct ratio from the correct angle's point of view.

  • Domain: Geometry and Trig
  • About 15% of the test is Geometry and Trig
  • Difficulty: easy to hard
  • Free, no account
Drill trigonometry now → See all Geometry and Trig skills

The patterns the SAT actually uses

Four patterns cover the trig you will meet. Each one is fast once you label the triangle from the named angle instead of from the picture's orientation.

Pattern 01

Complete the triple

Two sides given, third side wanted. The Pythagorean theorem always works, and recognizing the 3, 4, 5 and 5, 12, 13 families makes it instant.

Pattern 02

Pick the ratio

sin is opposite over hypotenuse, cos is adjacent over hypotenuse, tan is opposite over adjacent, all measured from the named angle, never from the right angle.

Pattern 03

Deploy the special triangles

The 45° 45° 90° triangle runs x, x, x√2. The 30° 60° 90° triangle runs x, x√3, 2x, with x opposite the 30° angle. Given any one side, scale the whole pattern.

Pattern 04

Swap sine and cosine

If A + B = 90°, then sin A = cos B. Questions that hand you one value and ask for the other are one line long once you spot the complement.

One worked example, start to finish

Worked example · medium

In right triangle ABC, angle C is the right angle and sin A = 5/13. What is the value of tan A?

  1. sin A is opposite over hypotenuse, so label the side opposite angle A as 5k and the hypotenuse as 13k.
  2. Find the third side with the Pythagorean theorem: (13k)² − (5k)² = 169k² − 25k² = 144k², so the adjacent side is √(144k²) = 12k.
  3. Build the ratio: tan A = opposite ÷ adjacent = 5k/12k = 5/12.
  4. Check the triple: 5² + 12² = 25 + 144 = 169 = 13². This is the 5, 12, 13 family, so every side is consistent.

Answer: tan A = 5/12

Every regenerated version is a triple wearing a different story. Label from the named angle and the ratios fall out.

Where students lose the point

  • Measuring from the wrong angle. Opposite and adjacent are defined by the angle in the question. The side opposite A is adjacent to B, and half the wrong choices are built from exactly that swap.
  • Swapping √2 and √3. The 45° 45° 90° triangle owns √2, on its hypotenuse. The 30° 60° 90° triangle owns √3, on its longer leg. Reversing them feels fine in the moment and fails.
  • Leaving the calculator in radians. Checking sin 30° in radian mode returns a negative decimal and instant panic. Confirm the angle mode before trusting any numeric check.
  • Treating the hypotenuse as a leg. The hypotenuse is always opposite the right angle and always the longest side. In a ladder or ramp story, identify it before assigning any ratio.

Using Desmos here

Desmos verifies trig answers fast if you watch the angle mode. From sin A = 5/13, compute arcsin(5 ÷ 13) in degrees to get about 22.6°, then check that tan(22.6°) returns about 0.4167, which is 5/12. Special triangle answers check the same way. Choosing the ratio is still your job, and leaving radian mode on is the classic way students hurt themselves here, so make the mode check a habit.

Why drilling here is different

Satified's trig drills regenerate sides, angles, and contexts endlessly, from quick triple completions to hard special triangle stacks, at easy, medium, and hard difficulty. Ratio picking turns into reflex because the numbers never repeat. Every answer in the bank of 1,483 questions has been independently verified.

New triangles until it is reflex.

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Questions students ask

Do I need the unit circle for the SAT?
Not deeply. The digital SAT leans on right triangle ratios, the two special triangles, and the complementary angle identity. Radians appear mostly in conversions and arc formulas, not as a full unit circle interrogation.
Are the special right triangle ratios provided?
Yes, both sit on the reference sheet: the 45° 45° 90° triangle with sides x, x, x√2, and the 30° 60° 90° triangle with sides x, x√3, 2x, where x is opposite the 30° angle. Memorizing them anyway turns those questions into 20 second wins.
What is the complementary angle identity?
In a right triangle the two acute angles add to 90°, so sin A = cos B whenever A + B = 90°. If a question gives sin 38° and asks for cos 52°, the answer is the same number, no calculator needed.
Will I need the law of sines or law of cosines?
No. SAT trigonometry stays inside right triangles: the Pythagorean theorem, the three basic ratios, special triangles, and similarity. If you are reaching for the law of cosines, there is a simpler intended path.
How does drilling this on Satified work?
Each question regenerates with fresh side lengths and angles at easy, medium, and hard difficulty, and every answer in the 1,483 question bank has been independently verified. You drill until picking the ratio is reflex, free and without an account.

Keep going

The right angle work here reappears inside circle questions, where tangents and radii build the very same triangles.