PSAT/NMSQT Math: Right Triangles and Trigonometry
Right triangle questions on the PSAT/NMSQT combine the Pythagorean theorem, the predictable ratios of special right triangles, and the three basic trigonometric ratios: sine, cosine, and tangent. This free, complete lesson covers the Pythagorean theorem, special right triangles, sine cosine and tangent, and complementary angle relationships. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quizzes below.
1. The Pythagorean Theorem
In any right triangle, the Pythagorean theorem relates the two legs (a and b) to the hypotenuse (c), the side opposite the right angle: a² + b² = c². It also works for related figures, like the diagonal of a square, which splits the square into two right triangles.
Worked Example: A right triangle has a hypotenuse of 17 and one leg of 8. Find the other leg.
8² + b² = 17² → 64 + b² = 289 → b² = 225.
b = √225 = 15
1. A right triangle has a hypotenuse of 17 and one leg of 8. Find the other leg. (see worked example above)
Explanation: 8² + b² = 17² → 64 + b² = 289 → b² = 225 → b = 15.
2. A square has a diagonal of 12√2. Find the length of one side.
Explanation: A square's diagonal splits it into a 45-45-90 right triangle, where the diagonal (hypotenuse) equals side × √2. So side = 12√2 / √2 = 12.
3. A right triangle has legs of 9 and 12. Find the length of the hypotenuse.
Explanation: 9² + 12² = c² → 81 + 144 = 225 → c = √225 = 15.
2. Special Right Triangles
Two right triangles show up so often that it pays to memorize their side ratios. A 45-45-90 triangle has legs of equal length x and a hypotenuse of x√2. A 30-60-90 triangle has a short leg x (opposite the 30° angle), a long leg x√3 (opposite the 60° angle), and a hypotenuse of 2x (opposite the 90° angle).
Worked Example: In a 30-60-90 triangle, the side opposite the 60° angle is 12. Find the side opposite the 30° angle.
The long leg (opposite 60°) equals x√3 = 12, so x = 12/√3 = 4√3.
The short leg (opposite 30°) equals x = 4√3
1. In a 30-60-90 triangle, the side opposite the 60° angle is 12. Find the side opposite the 30° angle. (see worked example above)
Explanation: The long leg (opposite 60°) = x√3 = 12, so x = 12/√3 = 4√3. The short leg (opposite 30°) equals x = 4√3.
2. An isosceles right triangle (45-45-90) has a hypotenuse of 8√2. Find the length of one leg.
Explanation: In a 45-45-90 triangle, hypotenuse = leg × √2. So leg = 8√2 / √2 = 8.
3. An equilateral triangle has sides of length 14. Find the length of its altitude (which splits it into two 30-60-90 triangles).
Explanation: The altitude splits the equilateral triangle into two 30-60-90 triangles with a short leg (half the base) of 7. The altitude is the long leg: 7√3.
3. Sine, Cosine, and Tangent
For an acute angle in a right triangle, the three basic trig ratios compare its sides: sin = oppositehypotenuse, cos = adjacenthypotenuse, and tan = oppositeadjacent. A few special angle values (like sin 30° = 1/2) are worth memorizing directly.
Worked Example: In a right triangle, angle A has an adjacent side of 6 and a hypotenuse of 6√3. If AC = 6√3 is the hypotenuse and AB = 6 is adjacent to a 60° angle, find cos(60°) using these measurements.
cos(60°) = adjacent/hypotenuse = 6 / 6√3 = 1/√3 = √3/3
1. In a right triangle, cos(60°) is found using an adjacent side of 6 and a hypotenuse of 6√3. What is cos(60°)? (see worked example above)
Explanation: cos(60°) = adjacent/hypotenuse = 6/(6√3) = 1/√3, which rationalizes to √3/3.
2. In a 45-45-90 right triangle, both legs measure 5 and the hypotenuse measures 5√2. Find sin(45°).
Explanation: sin(45°) = opposite/hypotenuse = 5/(5√2) = 1/√2, which rationalizes to √2/2.
3. In the same 45-45-90 right triangle (both legs = 5), find tan(45°).
Explanation: tan(45°) = opposite/adjacent = 5/5 = 1.
4. Complementary Angle Relationships
In a right triangle, the two acute angles are always complementary (they sum to 90°). This creates a powerful shortcut: sin(angle) = cos(90° − angle), for any acute angle. In other words, the sine of one acute angle always equals the cosine of the other.
Worked Example: In a right triangle, sin(35°) = 0.574. Find cos(55°).
Since 35° and 55° are complementary (35 + 55 = 90), sin(35°) = cos(55°).
cos(55°) = 0.574
1. In a right triangle, sin(35°) = 0.574. Find cos(55°). (see worked example above)
Explanation: The acute angles of a right triangle are complementary (sum to 90°). Since 35° + 55° = 90°, sin(35°) = cos(55°) = 0.574.
2. In a right triangle, one acute angle measures 42°. Find the measure of the other acute angle.
Explanation: The two acute angles of a right triangle sum to 90°: 90 − 42 = 48°.
3. If cos(x°) = sin(28°) for some acute angle x, what is the value of x?
Explanation: Since cos(x°) = sin(90° − x)°, setting 90 − x = 28 gives x = 62.
Common Mistakes to Avoid
- Mixing up which side is the hypotenuse in the Pythagorean theorem. The hypotenuse is always the longest side, opposite the right angle, and it's always isolated by itself (c²), never added with a leg.
- Forgetting the ratios for special right triangles, especially mixing up which leg is the "short leg" versus "long leg" in a 30-60-90 triangle relative to the given angle.
- Mislabeling opposite versus adjacent sides when setting up a sine, cosine, or tangent ratio. Always identify these relative to the specific angle being used, not a fixed side of the triangle.
- Forgetting that sine and cosine relate to different angles, not the same one. sin(θ) = cos(90° − θ) connects an angle to its complement, not to itself.
- Applying trig ratios to non-right triangles without adjustment. Sin, cos, and tan as defined here only apply directly inside a right triangle.
Frequently Asked Questions
Is there an easy way to remember sine, cosine, and tangent?
The mnemonic SOH-CAH-TOA is a common memory aid: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Why does sin(θ) always equal cos(90° − θ)?
In any right triangle, the side opposite one acute angle is the side adjacent to the other acute angle (since the two acute angles share the same two legs, just from opposite viewpoints). This swap is exactly why sine of one angle matches cosine of its complement.
Where can I practice more problems like these?
The Right Triangles and Trigonometry quizzes in the PSAT/NMSQT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.