 Hello and welcome to the session. In this session we discussed the following question which says, find the equation of tangent to the curve x equal to sin 3t, y is equal to cos 2t at t equal to pi upon 4. We know that equation of the tangent at the point x naught y naught to the curve y equal to fx is given by y minus y naught equal to f dash x naught into x minus x naught. This is the key idea to be used for this question. Now let's move on to the solution. Now the given curves are x equal to sin 3t, let this be equation 1 and we have y equal to cos 2t, let this be equation 2. Now in the next step we have differentiating the equations 1 and 2 with respect to t we get that is on differentiating this equation x equal to sin 3t we have dx upon dt is equal to 3 cos 3t. There are differentiating this equation y equal to cos 2t with respect to t we get dy by dt is equal to minus 2 sin 2t. Then next we have dy by dx is equal to dy by dt upon dx by dt that is this is equal to minus 2 sin 2t upon 3 cos 3t. Thus we get dy by dx is equal to minus 2 sin 2t upon 3 cos 3t. Next we need to find the slope of the tangent t equal to pi by 4 it is given by dy by dx at t equal to pi by 4. So this would be equal to minus 2 sin 2 into pi by 4 that is we put pi by 4 in place of t upon 3 cos 3 into pi by 4. This further is equal to minus 2 sin pi by 2 upon 3 cos 3 pi by 4. Now we know that sin pi by 2 is 1 so this is equal to minus 2 into 1 upon 3 into now we find out cos 3 pi by 4 this would be equal to cos of pi minus pi by 4 that is this would be equal to minus cos pi by 4 which gives us minus 1 upon root 2 is the value for cos 3 pi by 4. So minus 2 into 1 upon 3 into minus 1 upon root 2 so this would be equal to 2 into root 2 upon 3. So slope of the tangent at t equal to pi by 4 given by dy by dx at t equal to pi by 4 is equal to 2 root 2 by 3. Now let's find out the values for x and y when t is equal to pi by 4. So when we have t equal to pi by 4 then x would be equal to that is sin 3t and this would be equal to sin 3 pi by 4. Now sin 3 pi by 4 is equal to sin pi minus pi by 4 which is equal to sin pi by 4 and that is equal to 1 upon root 2. So we have sin 3 pi by 4 is 1 upon root 2. So we get when t is equal to pi by 4 x is equal to 1 upon root 2 and then when t is equal to pi by 4 then y is equal to cos 2t that is cos 2 into pi by 4 that is cos pi by 2 which is equal to 0 that is y is equal to 0 when we have t is equal to pi by 4. Hence we have equation of the tangent to the given curve at t equal to pi by 4 that is at the point 1 upon root 2 0 is given by y minus y naught that is 0 is equal to the slope of the tangent which is 2 root 2 upon 3. So we have y minus 0 is equal to 2 root 2 upon 3 into x minus x naught that is 1 upon root 2. So this implies we have 3 y is equal to 2 root 2 minus x naught 2 into x minus 1 upon root 2 that is we have 3 y is equal to 2 root 2 x minus 2 that is we have 2 root 2 x minus 3 y minus 2 is equal to 0 is the equation of the tangent. Thus our final answer is equation of tangent to the curve x equal to sin 3t y equal to cos 2t at t equal to pi by 4 is given by 2 root 2 x minus 3 y minus 2 equal to 0. So this completes the session. Hope you have understood the solution for this question.