How this instrument works
Inverse cosine (arccosine) answers the question 'what angle has this cosine value?' — given a ratio between an adjacent distance and a direct-line distance, arccosine returns the angle that ratio represents. This page frames that calculation specifically as a NAVIGATION problem: how far a course deviates from a reference direction, given how much progress was actually made along that direction versus the total distance traveled.
This is the identical mathematical operation as this site's more abstract Inverse Cosine (cos⁻¹) button-notation page, which works directly with a bare ratio rather than two labeled distances — the underlying arccosine function is the same either way, only the framing and the field labels differ, aimed at whichever audience finds one entry point more natural than the other.
The ratio must never exceed 1 (the adjacent distance can never be more than the direct-line distance) — arccosine is only defined for ratios between −1 and 1, since cosine itself never produces a value outside that range.
- Enter the adjacent distance (progress made along the reference direction) into the Adjacent distance field.
- Enter the direct-line distance actually traveled into the Direct-line distance field.
- Read Bearing angle: how far the course deviated from the reference direction.
- Try equal values for both distances to see the deviation drop to 0°.
Worked example — 3 miles adjacent, 5 miles direct
A ship travels 5 nautical miles in a direct line, covering 3 miles of eastward progress along the way — the course deviated from due east by acos(3⁄5)≈53.13°, found directly from the adjacent-to-hypotenuse ratio using inverse cosine.
A direct-line distance that exactly matches the adjacent distance (5 and 5) gives a deviation of exactly 0° — the course ran perfectly along the reference direction. An adjacent distance of 0 (no eastward progress at all) gives a deviation of exactly 90° — the course ran straight perpendicular to the reference line.
Questions
What does inverse cosine find here?
The angle a traveled course deviated from a reference direction, given how much of the total direct-line distance was actually progress ALONG that direction — a ratio of adjacent to hypotenuse, run backward through arccosine to recover the angle.
Why can't the adjacent distance exceed the direct-line distance?
Because the adjacent distance can never physically exceed the total distance traveled in a straight line — and mathematically, arccosine is only defined for ratios between −1 and 1, since cosine itself never produces a value outside that range.
How is this different from this site's plain Inverse Cosine (cos⁻¹) page?
Both compute the identical arccosine operation; this page frames it specifically as a navigation bearing question with labeled distances, while the other page works directly with a bare ratio using calculator-button notation.
What does a bearing of 0° mean?
That the course ran perfectly along the reference direction with no deviation at all — the adjacent distance exactly equals the direct-line distance traveled.
What does a bearing of 90° mean?
That the course ran entirely perpendicular to the reference direction, making zero progress along it — the adjacent distance dropped all the way to 0.