SAT Right triangles & trigonometry

Pythagorean theorem, SOH-CAH-TOA, and special triangles.

3% of MathMath · Geometry and Trigonometry4 question types
~1per test

How to score it

  • Pythagorean: a² + b² = c², with c the hypotenuse — know the 3-4-5 and 5-12-13 families.
  • In a right triangle, the two acute angles are complementary, so sin(A) = cos(B).
  • Special triangles' side ratios are on the reference sheet.

Common traps

  • sin(A) given, cos(A) asked — students forget the complementary identity sin(A) = cos(B).
  • Treats a leg as the hypotenuse.
  • Misapplies the 30-60-90 ratio order.

The 4 question types, with real examples

Pythagorean theorem

“What is the length of the [missing side]?”

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From the question bankMedium

In right triangle ABC, angle C is the right angle. If BC = 5 and AC = 12, what is the length of AB?

  • A77
  • B1313✓
  • C1717
  • D119119
Why B

The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse. Since BC = 5 and AC = 12 are the legs, AB2AB^{2} = 52+122=25+144=1695^{2} + 12^{2} = 25 + 144 = 169, so AB = 13.

sin / cos / tan in a right triangle

“If sin(A) = [fraction], what is cos(B) / tan(A)?”

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From the question bankMedium

Triangle DEF has a right angle at E. If DE = 8 and EF = 15, what is the value of sin D?

  • A817\displaystyle \frac{8}{17}
  • B158\displaystyle \frac{15}{8}
  • C815\displaystyle \frac{8}{15}
  • D1517\displaystyle \frac{15}{17}✓
Why D

First, use the Pythagorean theorem to find the hypotenuse DF: DF2DF^{2} = 82+152=64+225=2898^{2} + 15^{2} = 64 + 225 = 289, so DF = 17. The sine of angle D is the ratio of the opposite side to the hypotenuse. The side opposite angle D is EF = 15, so sin D=15/17D = 15/17.

Special 30-60-90 / 45-45-90 triangles

“What is the [hypotenuse / leg]?”

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From the question bankMedium

In right triangle MNO, angle N is the right angle. The measure of angle M is 45 degrees. If MN = 6, what is the length of MO?

  • A66
  • B62\displaystyle 6\sqrt{2}✓
  • C63\displaystyle 6\sqrt{3}
  • D1212
Why B

Since angle M is 45 degrees and angle N is 90 degrees, angle O must also be 45 degrees, making this a 45-45-90 triangle. In a 45-45-90 triangle, the sides are in the ratio 1 : 1 : 2\displaystyle \sqrt{2}. Since MN = 6 is one leg, the hypotenuse MO = 62\displaystyle 6\sqrt{2}.

Similar right triangles & trig values

“Triangle [FGH] is similar to triangle [JKL] (corresponding right angles). If sin(F) = [p/q], what is [sin(J) / cos(J)]?”

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From the question bankMedium

Triangle DEF is similar to triangle GHI, where angle D corresponds to angle G and angles E and H are right angles. If cos D=1213\displaystyle D = \frac{12}{13}, what is the value of cos G?

  • A513\displaystyle \frac{5}{13}
  • B512\displaystyle \frac{5}{12}
  • C1213\displaystyle \frac{12}{13}✓
  • D22\displaystyle \frac{\sqrt{2}}{2}
Why C

Since triangle DEF is similar to triangle GHI and angle D corresponds to angle G, these angles are congruent. Therefore, cos D = cos G=12/13G = 12/13.

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