# Question Video: Finding the Length of a Parametric Curve Mathematics • Higher Education

Find the length of the curve with parametric equations π₯ = 3 cos π‘ β cos 3π‘ and π¦ = 3 sin π‘ β sin 3π‘, where 0 β€ π‘ β€ π.

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### Video Transcript

Find the length of the curve with parametric equations π₯ equals three cos π‘ minus cos three π‘ and π¦ equals three sin π‘ minus sin three π‘, where π‘ is greater than or equal to zero and less than or equal to π.

We recall that the formula we used to find the arc length of a curve defined parametrically for values of π‘ from πΌ to π½ is the definite integral between πΌ and π½ of the square root of dπ₯ by dπ‘ squared plus dπ¦ by dπ‘ squared with respect to π‘. In this case, π₯ is equal to three cos π‘ minus cos three π‘ and π¦ is equal to three sin π‘ minus sin three π‘. And weβre interested in the length of the curve between π‘ is greater than or equal to zero and less than or equal to π.

So weβll let πΌ be equal to zero and π½ be equal to π. Weβre also going to need to work out dπ₯ by dπ‘ and dπ¦ by dπ‘. And so, since weβre working with trigonometric expressions, we recall the derivative of cos of ππ‘ and sin of ππ‘. They are negative π sin of ππ‘ and π cos ππ‘, respectively, for real constant values of π. This means dπ₯ by dπ‘ is negative three sin π‘ minus negative three sin three π‘. And of course, that becomes plus three sin three π‘. Similarly, dπ¦ by dπ‘ is three cos π‘ minus three cos three π‘.

Before we substitute into the formula, weβre actually going to square these and find their sum. Negative three sin π‘ plus three sin three π‘ all squared is nine sin squared π‘ minus 18 sin π‘ sin three π‘ plus nine sin square three π‘. Then, three cos π‘ minus three cos three π‘ all squared is nine cos squared π‘ minus 18 cos π‘ cos three π‘ plus nine cos squared three π‘. At this stage, we recall the trigonometric identity sin squared π‘ plus cos squared π‘ equals one. And we see that we have nine sin squared π‘ plus nine cos squared π‘. Well, that must be equal to nine. Similarly, we have nine sin squared three π‘ plus nine cos squared three π‘, which is also equal to nine. And we also have negative 18 times sin π‘ sin three π‘ plus cos π‘ cos three π‘. All Iβve done here is factored the negative 18 out.

Next, weβre going to use the trigonometric identity cos of π΄ minus π΅ is equal to cos π΄ cos π΅ plus sin π΄ sin π΅. And this means that sin π‘ sin three π‘ plus cos π‘ cos three π‘ must be equal to cos of three π‘ minus π‘, which is, of course, simply cos of two π‘. So this becomes 18 minus 18 cos of two π‘. And so, we find that the arc length is equal to the definite integral between zero and π of the square root of 18 minus 18 cos of two π‘ dπ‘.

Letβs clear some space and evaluate this integral. Now, in fact, the integral of the square root of 18 minus 18 cos of two π‘ still isnβt particularly nice to calculate. And so, we go back to the fact that cos of two π‘ is equal to two cos squared π‘ minus one. We replace cos of two π‘ with this expression and then distribute the parentheses. And our integrand is now equal to the square root of 36 minus 36 cos squared π‘. We take out the common factor of 36 and then rearrange the identity sin squared π‘ plus cos squared π‘ equals one. So that one minus cos squared π‘ is equal to sin squared π‘. So our integrand is six times the square root of sin squared π‘, which is, of course, simply six sin π‘.

When we integrate six sin π‘, we get negative six cos π‘. So the arc length is equal to negative six cos π‘ evaluated between those limits. Thatβs negative six cos of π minus negative six cos of zero, which is equal to 12. And so, we found that the arc length of the curve that weβre interested in is 12 units. As you might expect, not only does this process work for curves defined by trigonometric equations, but also those defined by exponential and logarithmic ones.