How do you find the equation of the tangent line to the curve at the given point #sin(x+y)=2x-2y# at point (pi, pi)?

Question

Elijah Abram · Accepted Answer

The equation is #y = 1/3x + (2pi)/3#.

Start by expanding #sin(x + y)# using the sum formula #sin(a + b) = sinacosb + sinbcosa# #sin(x + y) = 2x - 2y# #sinxcosy + sinycosx = 2x - 2y# Use a combination of implicit differentiation and the product rule: #cosxcosy - sinxsiny(dy/dx) + cosycosx(dy/dx) - sinysinx = 2 - 2(dy/dx)# #cosycosx(dy/dx) - sinxsiny(dy/dx) + 2(dy/dx) = 2 + sinxsiny - cosxcosy# #dy/dx(cosycosx - sinxsiny + 2) = 2 + sinxsiny - cosxcosy# #dy/dx = (2 + sinxsiny - cosxcosy)/(cosxcosy - sinxsiny + 2)# We can find the slope of the tangent by inserting our point, #(x, y)# into the derivative. Here, the point is #(pi, pi)# #m_"tangent" = (2 + 0(0) - (-1 xx -1))/(-1 xx -1 - 0 xx 0 + 2)# #m_"tangent" = 1/3# We can now use point-slope form to determine the equation of the tangent. #y - y_1 = m(x - x_1)# #y - pi = 1/3(x - pi)# #y - pi = 1/3x - pi/3# #y = 1/3x - pi/3 + pi# #y = 1/3x + (2pi)/3# So, the equation of the tangent is #y = 1/3x + (2pi)/3#. Hopefully this helps!

Ezekiel Alexander · Answer

undefined To find the equation of the tangent line to the curve at the given point (pi, pi), we need to find the derivative of the curve with respect to x. Taking the derivative of sin(x+y)=2x-2y with respect to x, we get cos(x+y) * (1+dy/dx) = 2.

Next, we substitute the coordinates of the given point (pi, pi) into the equation. Plugging in x=pi and y=pi, we have cos(pi+pi) * (1+dy/dx) = 2. Simplifying this equation gives cos(2pi) * (1+dy/dx) = 2. Since cos(2pi) = 1, the equation becomes 1 * (1+dy/dx) = 2.

Solving for dy/dx, we have 1 + dy/dx = 2. Subtracting 1 from both sides gives dy/dx = 1.

Therefore, the slope of the tangent line at the point (pi, pi) is 1. Using the point-slope form of a line, we can write the equation of the tangent line as y - pi = 1(x - pi), which simplifies to y = x.

Hence, the equation of the tangent line to the curve sin(x+y)=2x-2y at the point (pi, pi) is y = x.