Trigonometric ratios (trig ratios) define the relationship between the angles and side lengths of a right triangle. The three primary trig ratios are remembered using SOH-CAH-TOA: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent. These ratios are foundational in Florida MAFS Geometry standards (MAFS.912.G-SRT) and appear on the SAT Math “Additional Topics in Math” section.
Side opposite of angle / hypotenuse.
Formal definition: Trigonometric ratios are ratios of side lengths in a right triangle that correspond to a specific angle. Every right triangle has three sides relative to an acute angle: the opposite side (the side facing the angle), the adjacent side (the side next to the angle), and the hypotenuse (the longest side, always opposite the 90° angle). The three primary trigonometric ratios — sine, cosine, and tangent — each compare two of these three sides.
Where you’ll see it: Trig ratios appear in Florida MAFS Geometry standards (MAFS.912.G-SRT.C.6, C.7, C.8), Florida EOC Geometry assessments, SAT Math “Additional Topics in Math” (approximately 10% of the exam), ACT Mathematics (pre-calculus section), and are prerequisite knowledge for precalculus, AP Calculus, and AP Physics in Florida high schools.
SOH-CAH-TOA is the mnemonic every Florida geometry student learns to remember the three primary trigonometric ratios. Each syllable represents a ratio: SOH = Sine, CAH = Cosine, TOA = Tangent. Learn all three — the SAT tests them individually and in combination.
sin(θ) = O / H
Opposite ÷ Hypotenuse
cos(θ) = A / H
Adjacent ÷ Hypotenuse
tan(θ) = O / A
Opposite ÷ Adjacent
Use when: given the angle θ and need the ratio of the side opposite θ to the hypotenuse. On SAT: "In right triangle ABC, sin(A) = 3/5. What is cos(A)?" — requires knowing sin = O/H to identify O=3, H=5, then use Pythagorean theorem for A=4, so cos = 4/5.
Use when: given the angle θ and need the ratio of the side adjacent to θ to the hypotenuse. Key insight: sin(θ) = cos(90° - θ) — complementary angles swap sine and cosine. This relationship is a frequent SAT trap: "sin(30°) = cos(?°)" → cos(60°).
Use when: given the angle θ and need the ratio of the opposite to adjacent side. Key identity: tan(θ) = sin(θ) / cos(θ). Also: tan(θ) is undefined when cos(θ) = 0 (i.e., at 90°). Florida MAFS geometry problems frequently test tan for finding heights using angles of elevation.
Three examples covering the most common problem types: finding a missing side (Easy), finding a missing angle using inverse trig (Medium), and an SAT-style multi-step problem (Hard).
The trig ratios for 30°, 45°, and 60° are used so frequently in Florida geometry and on the SAT that they must be memorized — or derived quickly from the two special right triangles (30-60-90 and 45-45-90). The SAT does provide these triangles on its reference sheet, but knowing the values cold saves 30–60 seconds per question.
| Angle (θ) | Sin(θ) | Cos(θ) | Tan(θ) |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 = √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | Undefined |
| SAT Question Type | What It Tests | Frequency |
|---|---|---|
| Basic Trig Ratio | Given a right triangle with labeled sides, find sin/cos/tan of a given angle | 2–3 per test |
| Complementary Angles | sin(θ) = cos(90°–θ) — "sin(30°) = cos(?°)" format | 1–2 per test |
| Special Angle Values | Find exact value of sin/cos/tan for 30°, 45°, or 60° without a calculator | 1 per test |
| Multi-Step Trig + Pyth. | Use trig ratio to find one side, then Pythagorean theorem for another — or find a second ratio from a given one | 1 per test |
SOH-CAH-TOA is a mnemonic for the three primary trigonometric ratios in a right triangle: SOH = Sine equals Opposite over Hypotenuse (sin = O/H); CAH = Cosine equals Adjacent over Hypotenuse (cos = A/H); TOA = Tangent equals Opposite over Adjacent (tan = O/A). Each syllable represents one ratio using the first letter of each term in the formula.
The three sides of a right triangle are: the hypotenuse (the longest side, always opposite the 90° right angle), the opposite side (the side directly across from the reference angle), and the adjacent side (the side next to the reference angle that is not the hypotenuse). Important: “opposite” and “adjacent” change depending on which acute angle you are referencing.
Yes. Trig ratios (sine, cosine, tangent) appear in the SAT Math “Additional Topics in Math” domain, which accounts for approximately 10% of all SAT Math questions. The SAT does not provide the SOH-CAH-TOA definitions on its reference sheet — students must have these memorized. Common SAT trig question types include finding a missing side or angle, using the complementary angle identity (sin(θ) = cos(90°−θ)), and finding a second trig ratio given one ratio and a right triangle.
sin⁻¹(x) is the inverse sine function (also written arcsin) — it gives you the angle whose sine is x. For example, sin⁻¹(0.5) = 30° because sin(30°) = 0.5. csc(θ) is the cosecant — the reciprocal of sine: csc(θ) = 1/sin(θ) = Hypotenuse/Opposite. They are completely different operations. On SAT and Florida EOC exams, only sin⁻¹, cos⁻¹, and tan⁻¹ (inverse functions) are tested — not csc, sec, or cot.
Trigonometric ratios are covered in Florida MAFS Geometry standards: MAFS.912.G-SRT.C.6 (understand that by similarity, side ratios in right triangles define the trig ratios for acute angles), MAFS.912.G-SRT.C.7 (explain and use the relationship between the sine and cosine of complementary angles), and MAFS.912.G-SRT.C.8 (use trig ratios and the Pythagorean theorem to solve right triangle problems). These standards appear in Florida EOC Geometry assessments taken by students in Orlando, Winter Park, and Lake Nona.