Question

Find the derivatives of the following:

x = a (cos t + t sin t); y = a (sin t – t cos t)

Answer 1

x = a (cos t + t sin t), y = a (sin t – t cos t)

`("d"x)/("dt")` = a [– sin t + t cos t + sin t]

`("d"x)/("dt")` = at cos t  ........(1)

y = a (sin t – t cos t)

`("d"x)/("dt")` = a [cos t – (t × – sin t + cos t × 1)]

`("d"x)/("dt")` = a[cos t + t sin t – cos t]

`("d"x)/("dt")` = at sin t  .......(2)

From equations (1) and (2) we get

`(("d"y)/("dt"))/(("d"x)/("dt")) = ("at"sin"t")/("at" cos "t")`

`("d"y)/("d"x)` = tan t