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Right Triangles & Trigonometry for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about right triangles and trigonometry in one place: the Pythagorean theorem, special right triangles, SOH-CAH-TOA, the co-function identity, and angle of elevation/depression word problems — with step-by-step examples, worked problems, and 2 free practice quizzes.
Practice these quizzes
Free · No signupWhy right triangles matter
Right triangles anchor a large share of PSAT geometry, since many other shapes can be broken down into right triangles to find missing lengths and angles. Trigonometry on the PSAT stays intentionally narrow: it's built almost entirely from right triangles.
If a geometry problem feels stuck, look for a way to draw in a right triangle — a diagonal, a height, or a perpendicular segment. It's one of the most reliable problem-solving moves in PSAT geometry.
The Pythagorean theorem
A right triangle has legs of length 9 and 12. Find the hypotenuse.
Memorizing common Pythagorean triples — 3-4-5, 6-8-10, 9-12-15, 5-12-13 — lets you skip the calculation entirely when a triangle's sides are a multiple of one of these sets.
Special right triangles
| Triangle | Side ratio |
|---|---|
| 45°-45°-90° | leg : leg : hypotenuse = x : x : x√2 |
| 30°-60°-90° | short leg : long leg : hypotenuse = x : x√3 : 2x |
A 45-45-90 triangle has a leg of length 7. Find the hypotenuse.
In a 30-60-90 triangle, the side ratios are not evenly spaced multiples — the long leg involves √3, not just double the short leg.
SOH-CAH-TOA
In a right triangle, the side opposite angle θ is 5, and the hypotenuse is 13. Find sin(θ) and cos(θ) using the remaining side.
The co-function identity
In any right triangle, the two acute angles are complementary (they sum to 90°), creating a direct relationship between sine and cosine.
If sin(28°) = 0.469, what is cos(62°)?
Whenever two angles in a PSAT question sum to 90°, check whether the co-function identity lets you skip a calculation entirely.
Angle of elevation & depression
Angle of elevation
The angle measured upward from a horizontal line to an object above eye level.
Angle of depression
The angle measured downward from a horizontal line to an object below eye level.
A kite string makes a 50° angle of elevation with the ground. If the string is 120 feet long, how high is the kite? (Use sin(50°) ≈ 0.766)
Common PSAT traps
- Mixing up opposite and adjacent: these sides depend entirely on which angle you're referencing — re-identify them for each problem.
- Misapplying the 30-60-90 ratio: the long leg uses √3, and the hypotenuse is double the short leg — not the long leg.
- Forgetting the co-function identity applies only to complementary angles: the two angles must sum to exactly 90°.
- Confusing elevation and depression: both are measured from a horizontal line, in opposite vertical directions.
Test day strategy for right triangles & trigonometry
| Question signal | Fastest approach |
|---|---|
| Two sides of a right triangle given | Use the Pythagorean theorem, or check for a common triple first |
| 45° or 30°/60° angles mentioned | Apply the special right triangle ratios directly |
| One side and one angle given | Identify opposite/adjacent/hypotenuse, then apply SOH-CAH-TOA |
| Two angles summing to 90° | Apply the co-function identity to skip extra calculation |
| "Angle of elevation/depression" word problem | Sketch a right triangle with the horizontal as one leg |
Now put it to work
Two quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Right Triangles & Trigonometry 1
Pythagorean theorem, special right triangles, and SOH-CAH-TOA.
Start quiz → Quiz 2Right Triangles & Trigonometry 2
Co-function identity and angle of elevation/depression.
Start quiz →