Why Trigonometric Paths Need Care
A plane path described by sine and cosine can look simple, while its speed involves a square root that is not immediately easy to integrate. Trigonometric identities often reveal that the square root is a familiar expression, but signs and parameter intervals still matter. A missing absolute value or a repeated traversal can change the length.
For a position \(\mathbf{r}(t)=(x(t),y(t))\), differentiating the coordinates gives the velocity \(\mathbf{r}'(t)=(x'(t),y'(t))\). Its magnitude is the speed. If \(x\) and \(y\) are continuously differentiable on \([a,b]\), the length travelled on that parameter interval is the integral of speed.
This tutorial focuses on applying that formula to trigonometric coordinates. The main tasks are to differentiate correctly, simplify the speed without losing its sign information, and choose an interval that represents the intended trip along the curve.
A Useful Simplification Strategy
After differentiating, first combine the squares algebraically. Two identities are especially useful:
Other angle-sum identities can serve the same purpose. Once the expression under the square root is simplified, rewrite it as a square only if the equality is valid for every parameter value in the interval. For instance, \(\sqrt{\cos^2 t}=|\cos t|\), not \(\cos t\) in general.
Interval selection is a separate part of the calculation. A period tells you when the coordinates repeat, but that alone does not always establish that the curve is traced exactly once during that period. Check the coordinates and, when possible, show that one coordinate is strictly monotone. Strict monotonicity rules out a repeated point on the interval.
Worked Examples
Worked Example: A Path with a Half-Angle Identity
Find the length of \(\mathbf{r}(t)=(t+\sin t,-\cos t)\) for \(0\le t\le 2\pi\), and verify that this interval does not retrace part of the path.
Differentiate each coordinate:
Therefore the speed is
Using \(1+\cos t=2\cos^2(t/2)\), this becomes
The absolute value is necessary because \(\cos(t/2)\) changes sign between \(t=0\) and \(t=2\pi\). Thus
Split at \(u=\pi/2\), where cosine changes sign:
To check the interval, note that \(x'(t)=1+\cos t\ge 0\). For any \(a<b\) in \([0,2\pi]\), the integral \(\int_a^b(1+\cos t)\,dt\) is strictly positive: the continuous integrand is nonnegative and is not identically zero on any interval of positive length. Hence \(x(t)\) is strictly increasing, so the path cannot visit the same point twice. The length is \(8\).
Worked Example: A Higher-Frequency Coordinate Pair
Find the length of \(\mathbf{r}(t)=(t+\tfrac12\sin 2t,-\tfrac12\cos 2t)\) on \(0\le t\le\pi\), and check that this interval describes one non-retraced arc.
Differentiation gives
Then the speed simplifies as follows:
On \([0,\pi/2]\), cosine is nonnegative; on \([\pi/2,\pi]\), it is nonpositive. Therefore
For the interval check, \(x'(t)=1+\cos 2t=2\cos^2 t\ge0\). Its integral on any interval of positive length is positive, so \(x(t)\) is strictly increasing. The curve cannot retrace a point, and the stated interval gives one non-retraced arc. As a final derivative check, differentiating \(x\) and \(y\) recovers \(1+\cos 2t\) and \(\sin 2t\), whose squared sum is \(4\cos^2t\), as used in the speed calculation.
Worked Example: Combining Frequencies Before Taking the Square Root
For \(\mathbf{r}(t)=(\sin t+\tfrac12\sin 2t,\cos t-\tfrac12\cos 2t)\), compute the length on the full parameter interval \(0\le t\le2\pi\).
The coordinate derivatives are
Expand the squares and group the terms:
The third equality uses the cosine sum identity, and the last uses \(1+\cos 3t=2\cos^2(3t/2)\). Hence the speed is \(2|\cos(3t/2)|\), and
For the last integral, \(|\cos u|\) has period \(\pi\), and its integral over each interval of length \(\pi\) is \(2\). The transformed interval has length \(3\pi\), so it contains three such intervals. As a check on the speed calculation, squaring \(2|\cos(3t/2)|\) gives \(4\cos^2(3t/2)=2+2\cos3t\), which is exactly the squared velocity magnitude above.
The coordinates repeat after \(2\pi\), so this interval is a full parameter period. A full period is a natural interval for measuring the distance accumulated in one cycle of the parametrization. Do not infer from periodicity alone that every point on the geometric curve is visited only once; self-intersections or retracing must be checked separately when the question asks for one geometric traversal.
Why the Speed Formula Works Here
The trigonometric identities simplify the calculation, but the reason the resulting integral measures distance is the same as for any continuously differentiable plane path. Over a small parameter change \(\Delta t\), the position change is approximately \(\mathbf{r}'(t)\Delta t\). Its magnitude is approximately \(\|\mathbf{r}'(t)\||\Delta t|\), so the local distance per unit parameter is the speed. Adding these small distances over an interval leads to the integral of speed.
The same conclusion can be expressed using the arc-length function \(s(t)\), which records accumulated distance from a starting parameter \(a\). Where the curve is continuously differentiable, its rate of change is \(s'(t)=\sqrt{(x'(t))^2+(y'(t))^2}\). Integrating this rate from \(a\) to \(b\) gives the stated arc-length formula. Trigonometric identities do not alter the geometric formula; they make its integrand manageable.
Common Pitfall: Dropping the Absolute Value
In the first example, a tempting but incorrect step is
This is false whenever \(\cos(t/2)<0\). If used across the whole interval, it gives the wrong length:
A length of zero is impossible for this nonconstant path. The correction is \(\sqrt{4\cos^2(t/2)}=2|\cos(t/2)|\), followed by splitting the integral at the sign change. More generally, before removing an absolute value, identify the zeros of the trigonometric factor and determine its sign on each part of the interval.
Choosing an Interval for One Traversal
A period is a useful starting point, not a complete interval-selection argument. For a coordinate pair made from periodic terms, a common period returns both coordinates to their starting values. The parametrization may nevertheless repeat part of its path within that period. Conversely, a path with a term such as \(t\) in one coordinate need not be closed, even though other terms are periodic.
For the first two examples, monotonicity of \(x(t)\) proves that no parameter values in the interval give the same point: if \(x\) takes different values, the ordered pairs cannot be equal. For other curves, useful checks include comparing the coordinate pair at the proposed endpoints, looking for a smaller period, and testing whether the coordinates can agree at two distinct parameter values. State clearly whether the desired quantity is length over a specified parameter interval, distance accumulated during a full parameter cycle, or length of a geometric curve traced once.
Write down \(x'(t)\) and \(y'(t)\) before forming the speed.
Expand carefully, combine squares, and use identities only after the derivative components are correct.
Taking the square root of a square introduces an absolute value; split the interval if the sign changes.
Use periods, endpoint values, and monotonicity or another direct argument to detect repeated traversal.
Evaluate the nonnegative speed over the chosen interval and check that the result is consistent with the path and its interval.
Why These Calculations Matter
Trigonometric coordinates describe oscillating motion, periodic paths, and curves built from several frequencies. Their position formulas may be easy to write down, but distance depends on the combined effect of both coordinate changes. Simplifying the squared speed can turn a complicated-looking radical into a basic absolute-value integral, as the examples show.
The interval is part of the mathematical model, not a detail to add at the end. A correct speed integrated over an interval that omits part of a trip or repeats a section answers a different question. Keeping the algebra, sign analysis, and traversal check together makes the calculation both exact and interpretable.
Check Your Understanding
Work through these questions by differentiating first and checking the relevant signs and parameter intervals.
- For \(\mathbf{r}(t)=(2\sin t,2\cos t)\), find the speed. What is the length on \(0\le t\le\pi/2\)?
- Explain why \(\sqrt{(1-\cos t)^2}\) cannot always be replaced by \(1-\cos t\), and determine whether that replacement is valid for every real \(t\).
- For \(\mathbf{r}(t)=(t-\sin t,\cos t)\) on \(0\le t\le2\pi\), derive the speed and evaluate the length, showing any required split of the integral.
- What does a common period of the coordinate functions establish, and what additional issue must be checked before calling it one geometric traversal?
- Give a reason that strict increase of \(x(t)\) on an interval rules out a repeated point of the plane curve on that interval.
