SAT Math study guide · Chapter 22
Right Triangles and Trigonometry
The right triangle is where two of the domain's deepest tools live: the Pythagorean theorem ties its sides together, and trigonometry ties those sides to its angles.
This section combines several tested relationships in one figure: side lengths through the Pythagorean theorem, special-triangle ratios, and trigonometric ratios. The Pythagorean theorem relates its three sides, so any two give the third. Trigonometry goes further, linking the sides to the angles, so a side and an angle give everything. Layered on top are a few memorable shortcuts, the common triples and the two special triangles, that let you skip computation entirely when you recognize them. We build from the theorem outward to the trigonometry, and a great many geometry questions resolve, in the end, to spotting a right triangle and choosing the right tool.
▸ Apply the Pythagorean theorem
In any right triangle, the square of the hypotenuse (the side opposite the right angle, and always the longest) equals the sum of the squares of the two legs: a2 + b2 = c2, with c the hypotenuse. Given any two sides, this finds the third. Just keep c in its place: it is the hypotenuse alone, so when you solve for a leg you subtract.
FIGURE 1 The hypotenuse squared equals the sum of the squares of the legs. These three sides, 8-15-17, satisfy it exactly.
WORKED Find a side two ways
Find the hypotenuse of a right triangle with legs 6 and 8:
62 + 82 = 36 + 64 = 100 → c = 10.
Find a leg of a right triangle with hypotenuse 13 and one leg 5: now subtract, since you are solving for b:
b2 = 132 - 52 = 169 - 25 = 144 → b = 12.
▸ Recognize common Pythagorean triples
Some right triangles have whole-number sides, and recognizing these triples on sight saves all the squaring and square-rooting. Four are worth knowing cold, and because any multiple of a triple is itself a triple, each stands for a whole family.
Pythagorean triples worth recognizing
3-4-5, and its multiples 6-8-10, 9-12-15, 12-16-20, and so on.
5-12-13, and 10-24-26, and so on.
8-15-17 and 7-24-25.
Spot a multiple and you skip the arithmetic: legs 18 and 24 are 6 × (3-4-5), so the hypotenuse is 6 × 5 = 30.
▸ Use 45-45-90 and 30-60-90 side ratios
Two right triangles recur so often that their side ratios are printed on the reference sheet, and knowing them turns many problems into a single multiplication. The 45-45-90 triangle (an isosceles right triangle, the half of a square along its diagonal) has sides in the ratio 1 : 1 : √2: the two legs are equal, and the hypotenuse is a leg times √2. The 30-60-90 triangle, the half of an equilateral triangle down its altitude, has sides in the ratio 1 : √3 : 2, where the short leg faces the 30-degree angle, the long leg (short leg times √3) faces the 60, and the hypotenuse (twice the short leg) faces the right angle.
FIGURE 2 The two special right triangles and their side ratios: 1 : 1 : √2 for the 45-45-90, and 1 : √3 : 2 for the 30-60-90.
WORKED Use the special ratios
A square's diagonal: a square with side 6 splits into two 45-45-90 triangles, so its diagonal is a leg times √2, namely 6√2.
A 30-60-90: if the short leg (opposite the 30-degree angle) is 5, then the hypotenuse is 2 × 5 = 10 and the long leg is 5√3.
An equilateral altitude: the altitude of an equilateral triangle is the long leg of a 30-60-90 whose short leg is half the side, for side 10, the altitude is 5√3.
▸ Use sine, cosine, and tangent in right triangles
Trigonometry connects an acute angle of a right triangle to the ratios of its sides, packaged in the mnemonic SOHCAHTOA. For an acute angle θ, name the three sides by their role: the hypotenuse, the side opposite θ (across the triangle from it), and the side adjacent to θ (the leg touching it). Then the three ratios are sine = opposite ÷ hypotenuse, cosine = adjacent ÷ hypotenuse, and tangent = opposite ÷ adjacent.
FIGURE 3 Naming the sides relative to the angle θ gives the three ratios: SOH, CAH, TOA.
With a ratio and one side, you find another side; with two sides, you find the ratio, and from it the angle. On the test the angle is often one of the special values, so the special-triangle ratios and the trig ratios reinforce each other, sin 30° = 1/2 is just the 30-60-90 triangle read as a ratio.
WORKED Find a side with a ratio
A right triangle has an acute angle of 30 degrees and a hypotenuse of 14. Find the side opposite the 30-degree angle.
Sine is opposite over hypotenuse, and the sine of 30 degrees is 1/2, so the opposite side is half the hypotenuse:
opposite = 14 × (1/2) = 7.
If instead you are told sine of angle A is 3/5 and the hypotenuse is 20, the opposite side is 20 × (3/5) = 12.
▸ Use complementary-angle trig relationships
The two acute angles of a right triangle always sum to 90 degrees, they are complementary, and this produces an identity the test returns to often. The side opposite one acute angle is the side adjacent to the other, so the sine of one angle equals the cosine of the other. Written out: the sine of θ equals the cosine of (90 degrees − θ), and likewise the cosine of θ equals the sine of its complement.
FIGURE 4 The side opposite θ is adjacent to its complement, so the sine of θ equals the cosine of (90 degrees − θ).
WORKED Use the complementary identity
The sine of 40 degrees equals the cosine of 50 degrees, because 40 and 50 are complementary.
So if the sine of x degrees equals the cosine of 35 degrees, the two angles must be complementary:
x + 35 = 90 → x = 55.
This complementary relationship holds whenever both angles are acute, which covers every angle inside a right triangle. When both angles are acute, sin α = cos β implies α + β = 90 degrees, and setting the two angles' sum to 90 and solving is safe. If a problem ever allows a non-acute angle, do not assume complementarity without checking the stated domain first.
▸ Practice: right triangles and trigonometry
Where an answer is a radical such as 6√2, the grid-in versions accept its decimal value.
1. A right triangle has a hypotenuse of length 25 and one leg of length 7. What is the length of the other leg? (grid-in)
2. The two legs of a right triangle measure 18 and 24. What is the length of the hypotenuse? (grid-in)
3. A square has a diagonal of length 8√2. What is the length of each side of the square? (grid-in)
4. In a 30-60-90 triangle, the side opposite the 30-degree angle has length 5. What is the length of the hypotenuse? (grid-in)
5. An equilateral triangle has side length 10. Its altitude can be written as a√3. What is the value of a? (grid-in)
6. A right triangle has an acute angle of 30 degrees and a hypotenuse of length 14. What is the length of the side opposite the 30-degree angle? (grid-in)
7. If sin x° = cos 35°, where 0 < x < 90, what is the value of x? (grid-in)
8. In right triangle ABC, the right angle is at C, the sine of angle A is 3/5, and the hypotenuse AB has length 20. What is the length of side BC, which is opposite angle A? (grid-in)
9. In a right triangle, the tangent of one acute angle is 3/4, and the hypotenuse has length 20. What is the length of the shorter leg? (grid-in)
10. A rectangular box has dimensions 3 by 4 by 12. What is the length of the box's longest interior (space) diagonal? (grid-in)
11. A 13-foot ladder leans against a vertical wall with its base 5 feet from the wall. How high up the wall, in feet, does the ladder reach? (grid-in)
12. In a right triangle, the cosine of an acute angle θ is 0.6. What is the value of the sine of (90° − θ)? (grid-in)
13. In a 30-60-90 triangle, the longer leg (opposite the 60-degree angle) has length 6√3. What is the length of the shorter leg? (grid-in)
Question 14
- A. 6, 8, 9
- B. 5, 12, 13
- C. 7, 8, 11
- D. 9, 12, 16
15. A rectangle has a diagonal of length 13 and one side of length 5. What is the perimeter of the rectangle? (grid-in)
16. A ladder leans against a wall, making a 60° angle with the ground. The base of the ladder is 4 feet from the wall. To the nearest tenth of a foot, how high up the wall does the ladder reach? (grid-in)
17. Points A(1, 2), B(7, 2), and C(7, 10) lie in the coordinate plane. What is the length of AC? (grid-in)
▸ Solutions: right triangles and trigonometry
1: 24. Solve for the leg by subtracting: b2 = 252 - 72 = 625 - 49 = 576, so b = 24. These are a 7-24-25 triple.
2: 30. The legs are 6 × (3, 4) = (18, 24), so the hypotenuse is 6 × 5 = 30. Directly, √(182 + 242) = √900 = 30.
3: 8. In a 45-45-90 triangle the diagonal is the side times √2, so side = 8√2 ÷ √2 = 8.
4: 10. In a 30-60-90 triangle the hypotenuse is twice the short leg (the side opposite the 30-degree angle): 2 × 5 = 10.
5: 5. The altitude splits the equilateral triangle into two 30-60-90 triangles with short leg 10/2 = 5; the altitude is the long leg, 5√3. So a = 5.
6: 7. Sine is opposite over hypotenuse, and the sine of 30 degrees is 1/2, so the opposite side is 14 × (1/2) = 7.
7: 55. A sine equals a cosine when the angles are complementary, so x + 35 = 90 and x = 55.
8: 12. Sine of A is the opposite side over the hypotenuse: 3/5 = BC/20, so BC = 20 × (3/5) = 12.
9: 12. A tangent of 3/4 means the legs are 3k and 4k, so the hypotenuse is 5k. Then 5k = 20 gives k = 4, and the shorter leg is 3k = 12.
10: 13. The space diagonal extends the Pythagorean theorem to three dimensions: √(32 + 42 + 122) = √(9 + 16 + 144) = √169 = 13.
11: 12. The wall, ground, and ladder form a right triangle: √(132 - 52) = √(169 - 25) = √144 = 12 feet.
12: 0.6. The sine of an angle's complement equals its cosine, so the sine of (90° − θ) is exactly the cosine of θ, which is 0.6.
13: 6. The long leg is the short leg times √3, so the short leg is 6√3 ÷ √3 = 6.
Show the answers and explanations
14: B. Check a2 + b2 = c2: 52 + 122 = 25 + 144 = 169 = 132, so 5, 12, 13 works. The others do not satisfy the theorem.
15: 34. A rectangle’s diagonal splits it into two right triangles, with the diagonal as the hypotenuse and the two sides as legs. The missing side is √(13² - 5²) = √144 = 12, a 5-12-13 triple. The perimeter is 2(5 + 12) = 34. Nothing in the question mentions a triangle directly; recognizing that the diagonal creates one is the key step.
16: 6.9. The ladder, wall, and ground form a right triangle, with the height as the side opposite the 60° angle and the base as the adjacent side, so tangent connects them: tan 60° = h/4, giving h = 4√3 ≈ 6.9 feet. The scenario never uses the word triangle, but the ladder, wall, and ground are exactly its three sides.
17: 10. Points A and B share a y-coordinate, and B and C share an x-coordinate, so AB and BC are horizontal and vertical legs of a right triangle at B, with lengths 6 and 8. The hypotenuse AC is √(6² + 8²) = 10, a 6-8-10 triple, faster to spot than computing the distance formula directly.
▸ Right triangles and trigonometry: chapter summary
The right triangle answers to two tools. The Pythagorean theorem, a2 + b2 = c2, finds a third side from two (adding for the hypotenuse, subtracting for a leg) and the triples and the two special triangles let you skip that work when you recognize the pattern. Trigonometry then links sides to angles: sine is opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent, with the special angles giving clean values. And because a right triangle's acute angles are complementary, the sine of one is the cosine of the other. Find the right triangle in the figure, pick the tool that uses what you know, and the rest is arithmetic.