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Question
Find the derivatives of the following:
x = a (cos t + t sin t); y = a (sin t – t cos t)
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Solution
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
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