Right triangles and trigonometry
Geometry and Trigonometry is about 15% of SAT Math (5–7 of 44 questions), and right triangles are one of its main topics. Expect missing sides, special triangles, trigonometric ratios of an acute angle, and the link sin x = cos(90° − x). The reference sheet gives the Pythagorean relation and the special triangles.
The Pythagorean theorem
In a right triangle with legs a and b and hypotenuse c, a² + b² = c². To find a leg, subtract: a² = c² − b². Learn the common whole-number triples: they save time and the SAT uses them often. Multiples of a triple also work: 6, 8, 10 and 9, 12, 15 come from 3, 4, 5.
| Basic triple | Multiples |
|---|---|
| 3, 4, 5 | 6, 8, 10 · 9, 12, 15 · 30, 40, 50 |
| 5, 12, 13 | 10, 24, 26 · 15, 36, 39 |
| 8, 15, 17 | 16, 30, 34 |
| 7, 24, 25 | 14, 48, 50 |
The 45°-45°-90° triangle
This is half of a square. The two legs are equal, and the hypotenuse is the leg times √2. So if the leg is x, the hypotenuse is x√2. If you know the hypotenuse, divide by √2 to get the leg.
The 30°-60°-90° triangle
This is half of an equilateral triangle. The shortest leg, opposite the 30° angle, is x. The other leg, opposite the 60° angle, is x√3. The hypotenuse is 2x. Always find x, the short side, first.
Sine, cosine and tangent
For an acute angle, name the sides from the angle’s point of view: the opposite side faces the angle, the adjacent side touches it (and is not the hypotenuse). Then sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse and tan = opposite ÷ adjacent. The memory aid is SOH-CAH-TOA. The same triangle gives different ratios for its other acute angle, because opposite and adjacent switch.
Complementary angles and radians
The two acute angles of a right triangle add up to 90°. The side opposite one is adjacent to the other, so sin x° = cos(90 − x)°. For example, sin 20° = cos 70°. Angles can also be measured in radians: 180° = π radians, so multiply degrees by π/180 to get radians, and radians by 180/π to get degrees.
| Degrees | 30° | 45° | 60° | 90° | 180° | 360° |
|---|---|---|---|---|---|---|
| Radians | π/6 | π/4 | π/3 | π/2 | π | 2π |
- The diagonal is the hypotenuse of a right triangle with legs 24 and 18.
- 24² + 18² = 576 + 324 = 900.
- The diagonal is √900 = 30.
- Shortcut: 24 : 18 = 4 : 3, so this is the 3-4-5 triple multiplied by 6. The hypotenuse is 5 × 6 = 30.
- This is a 30°-60°-90° triangle. AC is the hypotenuse, so 2x = 20 and x = 10.
- BC is opposite the 30° angle, so BC = x = 10.
- The other leg is AB = x√3 = 10√3.
- 8/17
- 8/15
- 15/17
- 15/8
- cos A = adjacent ÷ hypotenuse.
- The side adjacent to A (touching A, not the hypotenuse) is AC = 8.
- The hypotenuse is AB = 17, so cos A = 8/17.
- Angles A and B are complementary: A + B = 90°.
- So cos B = cos(90° − A) = sin A.
- cos B = 0.8.
- Check with a triangle: sin A = BC/AB = 4/5, and cos B = BC/AB = 4/5, because BC is adjacent to B ✓
- Mixing up opposite and adjacentStart by putting your finger on the angle. Opposite is the side across from it; adjacent touches it. When you change to the other acute angle, the names switch.
- Adding the legsThe hypotenuse is √(a² + b²), not a + b. Check that your hypotenuse is the longest side but shorter than a + b.
- Wrong side for 30-60-90Short leg x (opposite 30°), long leg x√3 (opposite 60°), hypotenuse 2x. The longest side is always opposite the right angle.
- Using a calculator in the wrong modeIn Desmos, check whether the angle is in degrees or radians (the wrench icon). A wrong mode gives a wrong value.
- Forgetting the complementsin x = cos(90° − x). If a question has sin of one angle and cos of another, look for angles that add to 90°.
- Converting radians the wrong wayDegrees × π/180 → radians. Radians × 180/π → degrees. Check with 180° = π.
Desmos is a scientific calculator for trigonometry. Type sin(30) and Desmos answers in radians or degrees depending on the mode: open the wrench icon (Graph Settings) and choose degrees or radians. You can also solve for an unknown side: type the equation, such as 12/x = tan(35), and read the crossing point. A square root or a power is typed with sqrt( ) or ^.
- Open Graph Settings (the wrench) and choose Degrees
- Type sin(20) and cos(70): both show 0.3420…
- Type 3^2 + 4^2 to check a Pythagorean sum, or sqrt(24^2 + 18^2) to get a hypotenuse
- For radians, switch the mode and type sin(pi/6)
Use Desmos to check decimals of trig ratios or when the numbers are not a triple. Special triangles and triples are faster by hand.
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