The equation of the tangent to the curve x = t cost, y=t sin ⁡t at the origin, is

We have,

$\begin{array}{l} x = t \cos t and y = t \sin t \\ ∴ \frac{d x}{d i} = \cos t - t \sin t and \frac{d y}{d t} = \sin t + t \cos t \end{array}$

At the origin, we have

$x = 0, y = 0 \Rightarrow t \cos t = 0 and t \sin t = 0 \Rightarrow t = 0$

The slope of the tangent at $t = 0$ is

$\frac{d y}{d x} = {(\frac{d y}{\frac{d t}{d x}})}_{t = 0} = {(\frac{\sin t + t \cos t}{\cos t - t \sin t})}_{t = 0} = 0 .$

So, the equation of the tangent at the origin is

$y - 0 = (x - 0) \Rightarrow y = 0$ .

The equation of the tangent to the curve $x = t \cos t,$ $y = t \sin t$ at the origin, is