Definition
The inverse of the cosine function. If the cosine of an angle equals a given ratio, the arc cosine of that ratio returns the original angle. Written as cos⁻¹(x) or arccos(x).
Plain English
A math operation that works backwards from cosine. Cosine takes an angle and gives you a number; arc cosine takes that number and gives you back the angle.
Context Anchor
Seen in trigonometry-based navigation, avionics, engineering, and performance calculations where a formula needs to turn a ratio or computed value back into an angle.
Derivation
From Latin arcus, meaning 'bow' or 'arc.' In trigonometry, angles were originally measured along the arc of a circle, so 'arc cosine' literally means 'the arc (angle) whose cosine is this value.'
Why Pilots Care
Most pilots will not use arc cosine directly in normal cockpit work, but it can appear in aviation math, navigation formulas, avionics explanations, and technical training material. Knowing that it gives an angle helps prevent the formula from feeling mysterious.
Analogy
Cosine is like a machine that turns angles into ratios. Arc cosine is the same machine running in reverse — feed it the ratio, get the angle back.
Grounding Statement
Arc cosine turns a cosine value back into an angle.
Intuition Check
Do not read arc here as a curved flight path. In arc cosine, arc means the angle related to a value on a circle.
Example Sentence 1
To find the bank angle that produces a given horizontal force ratio, the technician used the arc cosine function on the calculator.
Example Sentence 2
After measuring the sides of the triangle on the chart, arc cosine gave the heading change needed for the crosswind.