### Video Transcript

Find the slope of the tangent line to the polar curve π equals cos two π at the point where π equals π by six.

Remember, to find the slope of the tangent line to a curve at a given point, we need to begin by calculating the derivative. For a polar curve of the form π equals π of π, itβs some function of π, dπ¦ by dπ₯ is equal to dπ¦ by dπ divided by dπ₯ by dπ. We use the formula dπ¦ by dπ equals dπ by dπ times sin π plus π cos π and dπ₯ by dπ equals dπ by dπ times cos π minus π sin π.

Now, if we go back to our question, we see that π is defined as cos of two π. And since our equations for dπ¦ by dπ and dπ₯ by dπ are in terms of dπ by dπ and π, weβre going to need to begin by calculating dπ by dπ. To do so, we quote the general result that the derivative of cos of ππ₯ is equal to negative π sin of ππ₯. And thatβs great, because that means dπ by dπ is negative two sin of two π.

Letβs now work out dπ¦ by dπ and dπ₯ by dπ. dπ¦ by dπ is dπ by dπ times sin π. So, thatβs negative two sin two π sin π plus π times cos π. And in this case, thatβs cos two π times cos π. We then use the formula for dπ₯ by dπ. Itβs dπ dπ times cos π. So, thatβs negative two sin two π times cos π. We then subtract π times sin π. So, here, we subtract cos two π sin π. dπ¦ by dπ₯ is the quotient of these. Itβs dπ¦ by dπ divided by dπ₯ by dπ. So, itβs negative two sin two π sin π plus cos two π cos π all over negative two sin two π cos π minus cos two π sin π.

Now, remember, weβre looking to work out the slope of the tangent line to our curve at the point where π is equal to π by six. Weβll achieve this by substituting π equals π by six into our expression for the derivative. And this gives us the expression shown. When we evaluate the numerator, we get negative root three over two plus root three over four. And the denominator, we get negative three over two minus one-quarter. The quotient of these is root three over seven. And so, we see the slope of the tangent line to the polar curve π equals cos two π at the point where π is equal to π by six is root three over seven.