Trigonometric Functions
Radian Measure, Degrees, Minutes and Seconds
The radian defined by arc length, converting between the two systems, and the sexagesimal subdivision of the degree.
What this page covers
- Define the radian from arc length and radius
- Convert between degrees and radians in both directions
- Convert between decimal degrees and degrees, minutes and seconds
- Recognise which system an angle is stated in
The radian
The angle subtended at the centre of a circle by an arc whose length equals the radius. The source states it as s = r, therefore θ = 1.
The definition makes the radian dimensionless: it is a ratio of two lengths, so the units cancel. That is exactly why it behaves so well in formulae.
A full turn traverses the whole circumference, s = 2pi r, so the angle is 2pi rr = 2pi radians. That single fact fixes the whole conversion.
Converting
The source derives the first factor rather than asserting it: from π radians = 180° it follows that 1° = π180 radians.
Degrees to radians — the source's examples
Leaving the answer as an exact multiple of π is preferable wherever it comes out neatly, as with 45°. A decimal is a last resort.
Radians to degrees — the source's example
The cancellation of π is the sign the conversion is the right way round. If π survives into an answer in degrees, the factor was inverted.
| Degrees | Radians | Degrees | Radians |
|---|---|---|---|
| 0° | 0 | 180° | π |
| 30° | π6 | 210° | 7pi6 |
| 45° | π4 | 270° | 3pi2 |
| 60° | π3 | 300° | 5pi3 |
| 90° | π2 | 360° | 2pi |
| 120° | 2pi3 | -90° | -π2 |
The source's rule is worth adopting: an angle written with a degree symbol is in degrees; one written as a bare number is in radians. So sin(2.3) means 2.3 radians, while sin(5.7°) means 5.7 degrees. A calculator in the wrong mode gives a plausible but wrong answer with no warning.
Degrees, minutes and seconds
Degrees subdivide in sixtieths, a convention inherited from Babylonian arithmetic. The source lays out the whole ladder.
| Unit | Symbol | Equals |
|---|---|---|
| Degree | ° | — |
| Minute | ' | 160 of a degree |
| Second | '' | 160 of a minute, 13600 of a degree |
| Third | ''' | 160 of a second (rarely used) |
DMS to decimal — the source's example
Convert 23° 30' 36'' to decimal degrees.
Decimal to DMS — the source's example
Convert 27.8396° to degrees, minutes and seconds.
- The whole degrees are 27. That leaves 0.8396°.
- Multiply the remainder by 60 to get minutes: 0.8396 × 60 = 50.376 minutes, so 50 whole minutes.
- Multiply the new remainder by 60 to get seconds: 0.376 × 60 = 22.56 seconds.
27 + 5060 + 22.63600 = 27 + 0.8333 + 0.00628 = 27.8396 ✓.
Standard position and coterminal angles
An angle is in standard position when its initial side lies along the positive x-axis and its vertex is at the origin. The terminal side is where it ends up.
- Initial side
- The positive x-axis
- Terminal side
- Where the rotation ends
- Positive angle
- Measured anticlockwise
- Negative angle
- Measured clockwise
Angles sharing the same terminal side. The source states that they are connected by multiples of 2pi.
Coterminal angles have identical values for every trigonometric function, since the functions depend only on the terminal point. This is periodicity stated in geometric language.
| Relationship | Condition | Example |
|---|---|---|
| Complementary | θ1 + θ2 = π2 (or 90°) | 30° and 60° |
| Supplementary | θ1 + θ2 = π (or 180°) | 50° and 130° |
Complementary angles matter because sin(π2 - θ) = cos θ — the 'co' in cosine means complement. The source derives this identity from the addition formula in Week 9.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| Multiplying by 180π to get radians | Multiply by π180 | π should survive, not cancel |
| Calculator in the wrong mode | Match the mode to the angle | sin 30 = 0.5 in degrees, -0.988 in radians |
| 1' = 1100° | 160° | The subdivision is sexagesimal |
| Using s = r θ with degrees | Radians only | The formula depends on the radian definition |
| Treating 2pi and 360° as different angles | They are the same rotation | Coterminal |
| Rounding a decimal degree before converting to DMS | Convert first, round last | Rounding early loses seconds |
Frequently asked questions
Why use radians at all?
Because arc length becomes simply s = rθ, with no conversion factor. Every formula involving rates of change of angle is cleaner in radians, which is why they become the default in later work.
How do I tell which system is meant?
By the degree symbol. The source's rule: sin(2.3) assumes radians, sin(5.7°) assumes degrees. An angle with no symbol is in radians.
Which way round is the conversion factor?
Multiply by π180 to go from degrees to radians, and by 180π to go the other way. If unsure, check against a known pair: 180° = π radians.
Are minutes and seconds still used?
Yes, in surveying, navigation and astronomy, where a fraction of a degree matters. One minute is 160 of a degree and one second is 160 of a minute.
Source. Handwritten teaching notes, Week 8, pages 1-3.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.