Trigonometry might sound like a subject best left in a high-school classroom, but for an Aviation Maintenance Technician (AMT), it is a practical workshop tool. Whether you are calculating the length of a diagonal brace, determining the mechanical advantage of a control cable routed over a pulley at an angle, or resolving the components of a force acting on a structural member, you will reach for sine, cosine, and tangent. The FAA AMT General written exam tests these concepts directly, and understanding them deeply — not just memorizing formulas — will help you apply them correctly under pressure.
This article builds trigonometry from the ground up, explains each of the three primary ratios, walks through worked examples relevant to aircraft maintenance, and highlights the exam traps that trip up the most test-takers.
Right Triangles: The Foundation
All basic trigonometry is built on the right triangle — a triangle containing one 90-degree angle. The three sides of a right triangle have specific names that depend on which of the two non-right angles (called acute angles) you are focused on at a given moment.
- Hypotenuse: Always the side directly across from the 90-degree angle. It is always the longest side of a right triangle.
- Opposite side: The side directly across from the acute angle you are working with (the reference angle).
- Adjacent side: The side that forms the reference angle along with the hypotenuse — in other words, the side next to the reference angle that is not the hypotenuse.
Note that opposite and adjacent are relative labels. If you shift your reference angle to the other acute angle in the same triangle, the opposite and adjacent sides swap roles. The hypotenuse never changes.
The Three Primary Trigonometric Ratios
A trigonometric ratio is simply a fraction formed by dividing one side of a right triangle by another. For a given reference angle, the three primary ratios are always constant — they depend only on the angle, not on the physical size of the triangle. This is a powerful fact: a tiny triangle and a massive structural assembly with the same angles will produce the same ratios.
Sine (sin)
The sine of an angle equals the length of the opposite side divided by the length of the hypotenuse.
sin(angle) = Opposite ÷ Hypotenuse
In a maintenance context, imagine a control cable that runs from a point on the fuselage wall diagonally to a bellcrank. If you know the cable's length (hypotenuse) and the angle it makes with the vertical wall, the sine of that angle tells you how far horizontally the cable reaches — in other words, the length of the opposite side.
Cosine (cos)
The cosine of an angle equals the length of the adjacent side divided by the length of the hypotenuse.
cos(angle) = Adjacent ÷ Hypotenuse
Using the same cable example: the cosine of the cable angle gives you the vertical component — the length of the adjacent side. This is critical when you need to know how much of the cable's tension is actually pulling in a useful direction versus pulling sideways on a bracket.
Tangent (tan)
The tangent of an angle equals the length of the opposite side divided by the length of the adjacent side. Note that the hypotenuse does not appear in this ratio at all.
tan(angle) = Opposite ÷ Adjacent
Tangent is particularly useful when you know two legs of a triangle and want to find the angle, or when you know the angle and one leg and need the other leg, without needing to involve the hypotenuse.
Memory Aid
The classic mnemonic for these three ratios is SOH-CAH-TOA:
- SOH — Sine = Opposite over Hypotenuse
- CAH — Cosine = Adjacent over Hypotenuse
- TOA — Tangent = Opposite over Adjacent
Say it out loud a few times and it sticks. Nearly every AMT student who masters trigonometry credits SOH-CAH-TOA as the anchor point from which all the work flows.
Using the Ratios: Worked Examples
The real skill is knowing which ratio to reach for. The key question is always: which two sides (or one side and the angle) are involved in my problem?
Example 1 — Finding an unknown side: A sheet-metal brace runs from a point on a rib to a spar. The brace makes a 35-degree angle with the spar, and the distance along the spar from the attachment point to directly below the rib attachment is 14 inches (the adjacent side). You want the length of the brace (hypotenuse).
You have the adjacent side and want the hypotenuse, so use cosine: cos(35°) = Adjacent ÷ Hypotenuse. Rearranging: Hypotenuse = Adjacent ÷ cos(35°). From a calculator or trig table, cos(35°) ≈ 0.819. Therefore: Hypotenuse = 14 ÷ 0.819 ≈ 17.1 inches. The brace must be cut to approximately 17.1 inches.
Example 2 — Finding an unknown angle: A rigging wire rises 9 inches vertically and spans 12 inches horizontally. What angle does it make with the horizontal?
You have the opposite (9 in) and adjacent (12 in), so use tangent: tan(angle) = 9 ÷ 12 = 0.75. To find the angle itself, use the inverse tangent function (arctan or tan⁻¹): angle = arctan(0.75) ≈ 36.9 degrees. The wire makes roughly a 37-degree angle with the horizontal.
Example 3 — Resolving forces: A tension load of 500 pounds acts along a cable at 40 degrees above horizontal. How much of that force acts horizontally, and how much acts vertically?
- Horizontal component (adjacent) = 500 × cos(40°) = 500 × 0.766 ≈ 383 lb
- Vertical component (opposite) = 500 × sin(40°) = 500 × 0.643 ≈ 321 lb
This kind of force resolution is fundamental to understanding loads on airframe structures and is tested on the AMT General exam.
Why Trigonometry Matters for AMTs
Aviation maintenance is a precision trade. A bracket drilled at the wrong angle, a cable rigged with incorrect geometry, or a structural repair with a misunderstood load path can compromise airworthiness. Trigonometry is not abstract — it is the math that ensures a diagonal member is cut to the correct length, that cable tension is translated accurately into control surface force, and that rivet patterns in angular structural repairs are properly laid out.
The FAA recognizes this by including trigonometry in the AMT General knowledge test. Mastery of these concepts also supports your work with the Pythagorean theorem (which lets you find the hypotenuse when both legs are known: c² = a² + b²) and with vectors, both of which appear in maintenance calculations and in further AMT study.
Key Numbers and Rules
- SOH-CAH-TOA: sin = Opp/Hyp, cos = Adj/Hyp, tan = Opp/Adj — these ratios are fixed for a given angle regardless of triangle size.
- Angle range: In a right triangle, each acute angle is between 0° and 90°. Sine and cosine values for these angles always fall between 0 and 1. Tangent values range from 0 to infinity.
- sin(30°) = 0.5, cos(60°) = 0.5: These two common angles share the same ratio value, a fact the exam may exploit to check whether you confuse sine and cosine.
- sin(45°) = cos(45°) ≈ 0.707: At 45°, sine and cosine are equal because the opposite and adjacent sides are equal lengths.
- Inverse functions: Use arcsin (sin⁻¹), arccos (cos⁻¹), or arctan (tan⁻¹) to work backwards from a ratio to find the angle.
- The three angles of any triangle sum to 180°. In a right triangle, the two acute angles therefore sum to 90°, meaning they are complementary.
- Rearranging the formula: If sin(A) = Opp/Hyp, then Opp = Hyp × sin(A) and Hyp = Opp ÷ sin(A). The same algebra applies to cosine and tangent.
Common Test Traps
- Mixing up opposite and adjacent: The most frequent mistake is forgetting that opposite and adjacent are defined relative to the reference angle you choose. Always label the triangle before picking your formula.
- Using sine when you need cosine (or vice versa): The exam often sets up problems where the intuitive-seeming ratio is the wrong one. Deliberately identify which two sides are involved before selecting SOH, CAH, or TOA.
- Forgetting to use the inverse function to find an angle: If you calculate a ratio (e.g., 0.75) and the question asks for the angle, you must apply arctan (or the appropriate inverse), not just report the decimal. Reporting 0.75 instead of approximately 36.9° is a classic error.
- Confusing the hypotenuse with a leg: The hypotenuse is always opposite the 90-degree angle and is always the longest side. Mislabeling a long leg as the hypotenuse corrupts every calculation that follows.
- Applying trig to non-right triangles without adjustment: SOH-CAH-TOA applies directly only to right triangles. If a problem presents an oblique (non-right) triangle, it must first be divided into right triangles or solved with the law of sines/cosines — a detail the AMT General exam may probe with a setup that tempts you to jump straight to basic trig on a non-right triangle.