Concept

Radian

The unit of angle equal to the arc length it cuts from a circle of radius one, so that a full turn is 2π. In radians the derivative of the sine is the cosine with no stray factor, which is why calculus uses them.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A circle unrolled into a sine wave. On the left a radius turns through an angle; on the right the height of its tip is plotted against the angle, tracing a sine curve.

A sine wave is a circle seen from the side

Sine is introduced as a ratio in a right triangle, which is true and explains nothing about why its graph is a wave. There is a better picture.

analysis · Circular functions

A circular sector of area 1.10 and a hyperbolic sector of area 0.80. On the left, the unit circle with the sector from (1, 0) to (cos 2.2, sin 2.2), of area 1.100. On the right, the hyperbola x² − y² = 1 with the sector from (1, 0) to (cosh 1.6, sinh 1.6), of area 0.800. In both, the parameter is twice the shaded area.

The angle that is really an area

On a unit circle the angle t is the length of arc the point has walked, and it is also twice the area of the slice it has swept. The two readings agree on the circle and part company on the hyperbola — where arc length leads nowhere and area leads straight to cosh, sinh, and the exponential.

analysis · Circular functions

Named alongside it

The objects these essays reach for when they reach for this one.

Cosine Sine Unit circle Area Derivative Exponential Hyperbola Parametrisation Periodicity Pi Simple harmonic motion

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