# Why is using the arc length of a circle, $s$, equivalent to using the angle measure in radians, $\theta$, as the input of the trigonometric functions?

Upon introducing the idea of circular functions my textbook states:

We start at the point $$(1,0)$$ and measure an arc length $$s$$ along the circle. [...] Let the endpoint of this arc be at the point $$(x, y)$$. The circle is the unit circle - it has center at the origin and radius 1 unit (hence the name unit circle). Recall from algebra that the equation of this circle is: $$x^2 + y^2 =1$$

The radian measure of $$\theta$$ is related to the arc length $$s$$. For $$\theta$$ measured in radians, we know that $$s=r\theta$$. Here $$r=1$$, so $$s$$, which is measured in linear units such as inches or centimeters, is equal to $$\theta$$, measured in radians. Thus, the trigonometric functions of angle $$\theta$$ in radians found by choosing a point $$(x, y)$$ on the unit circle can be rewritten as functions of the arc length $$s$$, a real number. When interpreted this way, they are called circular functions. I have several questions regarding this:

1) The text points out that the arc length, $$s$$ is measured in linear units while the angle measure, $$\theta$$, is in radians. Clearly two different units of measurement, so how could the two unit types be used interchangeably as the inputs of the trigonometric functions, where, for example, $$\sin(s)= \sin(\theta)$$?

2) $$s=\theta$$ is derived from $$s=r\theta$$ (formula for calculating arc length) and the fact that a unit circle has a radius of $$1$$, $$r=1$$. However, if the unit circle is not used and the radius is no longer 1, then is the arc length, $$s$$, still equal to the angle measure $$\theta$$? How?

3) Why not continue using angle measure in radians as the input for the trig. functions? Why use arc length instead?

• FYI: This answer illustrates how the sine and cosine values arise as power series of the arc-length value. – Blue Mar 5 '19 at 2:38

The radian measure of an angle is $$\frac{\text{length of arc}}{\text{radius of circle}}$$ which is dimensionless.
A central angle of $$\theta$$ radians in a circle of radius $$r$$ will cut off an arc of length $$r \theta$$. The $$\sin$$ of that angle is independent of $$r$$.
In the figure the circle has radius $$1$$, so the radian measure of the angle is numerically equal to the arclength. But it is the radian measure that is the argument you use when computing the $$\sin$$.
• If the radian measure is the argument for the trig functions, then why bother using the arc length as the argument and going through the point of showing that $s = \theta$? – Slecker Mar 5 '19 at 2:58
• You don't use the arc length, unless you happen to have or want to construct a unit circle on which to measure the arc length. I think the illustration in your book is subtly wrong. The point is $(\cos \theta, \sin \theta )$. Most students won't notice, but you read carefully enough to ask this good question. – Ethan Bolker Mar 5 '19 at 3:05