Right Triangles and Trigonometry: Free SAT Math Lesson
Right triangles and trigonometry give you tools that work together: the Pythagorean theorem and special right triangles let you find missing side lengths, while the trigonometric ratios connect side lengths to angle measures. This free, complete lesson covers the Pythagorean theorem, the 45-45-90 and 30-60-90 special right triangles, the sine, cosine, and tangent ratios (SOH-CAH-TOA), the complementary angle identity, and angle-of-elevation word problems. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.
1. The Pythagorean Theorem
For any right triangle with legs a and b and hypotenuse c (the side opposite the right angle):
a² + b² = c²
a² + b² = c², where c is the hypotenuse
Worked Example: A right triangle has legs 9 and 12. Find the hypotenuse.
9² + 12² = 81 + 144 = 225
c = √225 = 15
1. A right triangle has legs 5 and 12. What is the length of the hypotenuse?
Explanation: 5² + 12² = 25 + 144 = 169, and √169 = 13.
2. A right triangle has a hypotenuse of 25 and one leg of 7. What is the length of the other leg?
Explanation: 25² − 7² = 625 − 49 = 576, and √576 = 24.
3. A right triangle has legs of 8 and 15. What is the length of the hypotenuse?
Explanation: 8² + 15² = 64 + 225 = 289, and √289 = 17.
4. A ladder 13 feet long leans against a wall, with its base 5 feet from the wall. How high up the wall does the ladder reach?
Explanation: The ladder is the hypotenuse: 13² − 5² = 169 − 25 = 144, and √144 = 12.
5. Which set of side lengths could NOT form a right triangle?
Explanation: For 7, 10, 13: 7² + 10² = 49 + 100 = 149, but 13² = 169. Since 149 ≠ 169, this is not a right triangle. The other three sets are all valid Pythagorean triples.
2. Special Right Triangles
Two right triangles come up often enough to memorize their exact side ratios directly, without needing the Pythagorean theorem each time.
45-45-90 triangle – legs equal, hypotenuse = x√2
30-60-90 triangle – sides x, x√3, and 2x
Worked Example: A 45-45-90 triangle has a leg of length 8. Find the hypotenuse.
The hypotenuse of a 45-45-90 triangle is always the leg length times √2.
hypotenuse = 8√2
1. A 45-45-90 triangle has a leg of length 6. What is the length of the hypotenuse?
Explanation: The hypotenuse is the leg times √2: 6√2.
2. A 45-45-90 triangle has a hypotenuse of length 10√2. What is the length of each leg?
Explanation: Divide the hypotenuse by √2: 10√2√2 = 10.
3. A 30-60-90 triangle has a short leg (opposite the 30° angle) of length 5. What is the length of the hypotenuse?
Explanation: The hypotenuse is always twice the short leg: 2(5) = 10.
4. A 30-60-90 triangle has a short leg of length 5. What is the length of the long leg (opposite the 60° angle)?
Explanation: The long leg is always the short leg times √3: 5√3.
5. A 30-60-90 triangle has a hypotenuse of length 14. What is the length of the short leg?
Explanation: The short leg is half the hypotenuse: 14 ÷ 2 = 7.
3. Trigonometric Ratios (SOH-CAH-TOA)
For an acute angle θ in a right triangle, the three basic trigonometric ratios compare pairs of sides:
| Ratio | Definition | Memory aid |
|---|---|---|
| sin(θ) | oppositehypotenuse | SOH |
| cos(θ) | adjacenthypotenuse | CAH |
| tan(θ) | oppositeadjacent | TOA |
Worked Example: In a right triangle, angle θ has an opposite side of 6, an adjacent side of 8, and a hypotenuse of 10. Find sin(θ), cos(θ), and tan(θ).
sin(θ) = 610 = 35 cos(θ) = 810 = 45 tan(θ) = 68 = 34
1. In a right triangle, angle θ has an opposite side of 9 and a hypotenuse of 15. What is sin(θ)?
Explanation: sin(θ) = oppositehypotenuse = 915 = 35.
2. In a right triangle, angle θ has an adjacent side of 12 and a hypotenuse of 13. What is cos(θ)?
Explanation: cos(θ) = adjacenthypotenuse = 1213.
3. In a right triangle, angle θ has an opposite side of 7 and an adjacent side of 24. What is tan(θ)?
Explanation: tan(θ) = oppositeadjacent = 724.
4. A right triangle has legs 6 and 8, with hypotenuse 10. For the angle opposite the side of length 6, what is sin(θ)?
Explanation: sin(θ) = oppositehypotenuse = 610 = 35.
5. Using the same triangle as the previous problem (legs 6 and 8, hypotenuse 10), what is cos(θ) for that same angle?
Explanation: The side adjacent to this angle is the other leg, 8. cos(θ) = adjacenthypotenuse = 810 = 45.
4. Using Trigonometry to Solve Problems
In any right triangle, the two acute angles are complementary (they sum to 90°), which leads to a useful identity:
sin(θ) = cos(90° − θ)
Trigonometry also solves real-world problems involving an angle of elevation (measured upward from horizontal) or angle of depression (measured downward from horizontal).
Worked Example: If sin(35°) = cos(x°), what is the value of x?
Using the identity sin(θ) = cos(90° − θ):
x = 90 − 35 = 55
1. If sin(28°) = cos(x°), what is the value of x?
Explanation: x = 90 − 28 = 62.
2. A 20-foot ladder leans against a wall, making a 60° angle with the ground. How high up the wall does the ladder reach, to the nearest foot? (sin 60° ≈ 0.866)
Explanation: The height is opposite the 60° angle, and the ladder is the hypotenuse: height = 20 × sin(60°) ≈ 20(0.866) ≈ 17.3, which rounds to 17 feet.
3. A ramp rises 3 feet over a horizontal distance of 12 feet. What is the angle of elevation of the ramp, to the nearest degree? (tan⁻¹(0.25) ≈ 14°)
Explanation: tan(θ) = riserun = 312 = 0.25, and tan⁻¹(0.25) ≈ 14°.
4. If cos(x°) = sin(50°), what is the value of x?
Explanation: Using the identity, x = 90 − 50 = 40.
Common Mistakes to Avoid
- Mixing up which side is opposite versus adjacent. "Opposite" and "adjacent" are always defined relative to a specific angle, not fixed positions in the triangle. Relabel carefully whenever the angle in question changes.
- Applying the Pythagorean theorem to a non-right triangle. a² + b² = c² only holds for right triangles. Confirm the right angle before using it.
- Forgetting that the hypotenuse is always the longest side. In a² + b² = c², c must represent the hypotenuse. Plugging a given side into the wrong position in the formula gives an incorrect, sometimes impossible, result.
- Mixing up the 45-45-90 and 30-60-90 side ratios. A 45-45-90 triangle has two equal legs and a hypotenuse of leg × √2. A 30-60-90 triangle has three different side lengths in the ratio x : x√3 : 2x. Confusing which ratio applies to which triangle is a very common error.
- Using degrees and radians inconsistently. Make sure your calculator or reasoning matches the units given in the problem, an angle problem stated in degrees should be solved in degree mode.
Frequently Asked Questions
Do I need a calculator with trig functions for the Digital SAT?
The Digital SAT allows an approved graphing calculator throughout the entire Math section, and many trigonometry questions are designed to be solved with one. Even so, understanding the underlying ratios and special triangle relationships helps you work faster and catch calculator entry errors.
Why are the 45-45-90 and 30-60-90 triangles worth memorizing?
These two triangles appear constantly, and their side ratios come directly from geometric reasoning (a 45-45-90 triangle is half of a square cut along its diagonal, and a 30-60-90 triangle is half of an equilateral triangle cut through its height). Knowing the ratios by heart lets you skip the Pythagorean theorem entirely for these specific triangles.
What's the difference between angle of elevation and angle of depression?
Angle of elevation is measured upward from a horizontal line, like looking up at the top of a building. Angle of depression is measured downward from a horizontal line, like looking down from a cliff at something below. The two angles are equal whenever they connect the same two points, by the property of alternate interior angles with parallel horizontal lines.
Where can I practice more problems like these?
The Right Triangles & Trigonometry quizzes in the SAT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill on the Digital SAT.