Find the slope of the tangent line to 𝑟 equals 𝜃 at 𝜃 equals 𝜋 by two.
We’ve been given an equation to a polar curve, 𝑟 is equal to some function of 𝜃. And we’re asked to find the slope of the tangent line to the curve at a given point, where 𝜃 is equal to 𝜋 by two. To find the slope, we need to find the derivative of 𝑦 with respect to 𝑥. And so, we use a special formula. This formula says that d𝑦 by d𝑥 is equal to d𝑟 by d𝜃 sin 𝜃 plus 𝑟 cos 𝜃 all over d𝑟 by d𝜃 cos 𝜃 minus 𝑟 sin 𝜃. We can see we’re going to need to begin then by working out d𝑟 by d𝜃. 𝑟 is equal to 𝜃, and the derivative of 𝜃 with respect to 𝜃 is simply one. So we say that d𝑟 by d𝜃 is equal to one. Replacing d𝑟 by d𝜃 in our equation for the slope, we get one sin 𝜃 plus 𝑟 cos 𝜃 over one cos 𝜃 minus 𝑟 sin 𝜃. And, of course, we don’t really need these ones.
But we’re also told that 𝑟 is equal to 𝜃. So let’s replace 𝑟 with 𝜃 and we have an equation for d𝑦 by d𝑥 purely in terms of one variable, in terms of 𝜃. The slope of the tangent line to 𝑟 equals 𝜃 is given by the value of d𝑦 by d𝑥 at the point where 𝜃 is equal to 𝜋 by two. So we substitute 𝜃 equals 𝜋 by two into sin 𝜃 plus 𝜃 cos 𝜃 over cos 𝜃 minus 𝜃 sin 𝜃. Now, of course, sin of 𝜋 by two is one. Cos of 𝜋 by two is zero. And we repeat these values on the denominator of our fraction. So we get one plus zero as our numerator. And our denominator is zero minus 𝜋 by two, which simplifies to one over or divided by negative 𝜋 by two.
Now, of course, to divide by a fraction, we multiply by the reciprocal of that fraction. So d𝑦 by d𝑥 evaluated at 𝜃 equals 𝜋 by two is one multiplied by negative two over 𝜋, which is simply negative two over 𝜋. And so, the slope of the tangent line to 𝑟 equals 𝜃 at 𝜃 equals 𝜋 by two is negative two over 𝜋.