Right Triangles and Trigonometry: SAT Practice Questions & Study Guide
Questions applying the Pythagorean theorem, SOH-CAH-TOA, and complementary angle trig identities to find side lengths and angle measures in right triangles.
Understanding Right Triangles and Trigonometry on the SAT
The Pythagorean theorem states that in any right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle). Pythagorean triples are integer solutions to this equation; the most common on the SAT are 3-4-5, 5-12-13, and 8-15-17, along with their multiples (e.g., 6-8-10 and 9-12-15). Recognizing these triples on sight saves you from doing the square root every time.
Trigonometric ratios define the relationships between angles and sides in a right triangle. For an acute angle θ: sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent. The mnemonic SOH-CAH-TOA encodes these. The critical habit is always to label 'opposite' and 'adjacent' relative to the specific angle being used—these labels are not fixed; they depend on which angle you're referencing.
Complementary angle trig identities follow from the fact that the two acute angles in a right triangle sum to 90°. This means sinθ = cos(90° − θ) and cosθ = sin(90° − θ). Equivalently, if sinA = cosB in a right triangle, then A + B = 90°. This relationship appears frequently on the Digital SAT in the form 'if sin(x°) = cos(y°), what is x + y?' The answer is 90 whenever x and y are acute angles.
Inverse trig functions (arcsin, arccos, arctan) allow you to find angle measures when you know the ratio of two sides. On the Digital SAT, a question might give you two sides of a right triangle and ask for a specific angle measure—you use the appropriate inverse trig function on your calculator. Radian measure appears occasionally: π radians = 180°, so 1 radian ≈ 57.3°. Key conversions to know: π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°.
Key Rules & Formulas
Memorize these rules — they come up directly in SAT questions.
Pythagorean theorem: a² + b² = c² for legs a, b and hypotenuse c
Legs 5 and 12: hypotenuse = √(25 + 144) = √169 = 13
SOH-CAH-TOA: sinθ = opp/hyp; cosθ = adj/hyp; tanθ = opp/adj
In a right triangle with opposite = 3 and hypotenuse = 5: sinθ = 3/5 = 0.6, so θ ≈ 36.87°
Complementary angle identity: sinθ = cos(90° − θ); if sinA = cosB then A + B = 90°
sin(32°) = cos(58°); if sin(x°) = cos(40°), then x + 40 = 90, so x = 50
Radian–degree conversion: 180° = π radians; to convert degrees to radians, multiply by π/180
60° × (π/180) = π/3 radians; π/4 radians × (180/π) = 45°
Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17 (and multiples)
A right triangle with legs 9 and 12 has hypotenuse 15 (3× the 3-4-5 triple)
Right Triangles and Trigonometry Practice Questions
Select an answer and click Check Answer to reveal the full explanation. Questions go from easiest to hardest.
In right triangle ABC, angle B = 90°, leg AB = 6, and leg BC = 8. What is the length of hypotenuse AC?
In right triangle PQR, angle R = 90°. If angle P = 30° and the hypotenuse PQ = 20, what is the length of the leg QR (opposite angle P)?
In right triangle ABC with angle C = 90°, sin(A) = 3/5. What is cos(A)?
In right triangle DEF, angle F = 90°, angle D = 38°, and DE = 15 (hypotenuse). What is the length of leg EF (opposite angle D) to the nearest tenth? (sin 38° ≈ 0.616)
If sin(x°) = cos(34°), what is the value of x?
A 26-foot ladder leans against a wall. The base of the ladder is 10 feet from the wall. How high up the wall does the ladder reach?
An isosceles right triangle has legs of length 9. What is the length of the hypotenuse?
In right triangle ABC, angle B = 90°. tan(A) = 5/12. If AC (the hypotenuse) = 26, what is the length of AB (adjacent to angle A)?
A surveyor stands 80 meters from the base of a building and measures the angle of elevation to the top of the building as 52°. What is the height of the building to the nearest meter? (tan 52° ≈ 1.280)
In right triangle RST, angle T = 90°. The measure of angle R is (4x + 10)° and the measure of angle S is (5x − 10)°. What is the value of sin(R)?
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Common Mistakes to Avoid
These are the most frequent errors students make on Right Triangles and Trigonometry questions. Knowing them in advance prevents costly point losses.
- !Labeling 'opposite' and 'adjacent' relative to the right angle instead of the angle θ being used
- !Using sin when the problem requires cos, or vice versa, because the labels weren't set relative to the correct angle
- !Forgetting to take the square root at the end of the Pythagorean theorem, leaving the answer as c² instead of c
- !Adding or subtracting 90° instead of subtracting from 90° when applying the complementary angle identity
- !Confusing radians and degrees on the calculator—make sure the calculator is in the correct mode (degrees vs. radians) before computing a trig value
SAT Strategy Tips: Right Triangles and Trigonometry
Before writing any trig ratio, label all three sides of the right triangle relative to the angle in the question—never proceed without this step
When you spot a right triangle with integer sides, check for a Pythagorean triple before computing—it's usually faster
For complementary angle questions, the fastest method is to notice that 'sinA = cosB' means A + B = 90° and solve directly by adding
Make sure your calculator is in degree mode for SAT trig problems unless the problem explicitly gives angles in radians
Other Geometry & Trigonometry Subtopics
Area and Volume
Questions asking you to compute or compare areas of 2D figures and surface areas and volumes of 3D solids.
Lines, Angles, and Triangles
Questions covering angle relationships formed by parallel lines, properties of triangles, congruence, similarity, and the triangle inequality.
Circles
Questions covering circle area, circumference, arc length, sector area, central and inscribed angles, and the equation of a circle in the coordinate plane.
Master Right Triangles and Trigonometry on the SAT
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