Find an equation of the tangent line at the point (𝜋/4,𝜋/4)
sin(𝑥−𝑦)=𝑥*cos(𝑦+𝜋/4)
sin(x-y) = x cos(y + π/4)
cos(x-y) - cos(x-y)y' = cos(y + π/4) - x sin(y + π/4) y'
y' = [cos(y+ π/4) - cos(x-y)] / [x sin(y + π/4) - cos(x-y)]
at (π/4,π/4), that means
y' = [0 - 1] / [π/4 - 1] = 1/(1 - π/4)
Now you have a point and a slope, so the line is
y - π/4 = 1/(1 - π/4) (x - π/4)
see the graphs at
https://www.wolframalpha.com/input/?i=plot+sin%28x-y%29+%3D+x+cos%28y+%2B+%CF%80%2F4%29%2C+y+-+%CF%80%2F4+%3D+1%2F%281+-+%CF%80%2F4%29+%28x+-+%CF%80%2F4%29+for+0+%3C%3D+x+%3C%3D++%CF%80%2F3%2C+0+%3C%3D+y+%3C%3D++%CF%80%2F3
To find the equation of the tangent line at a point in a given curve, you need to find the derivative of the curve and evaluate it at the point of interest. Let's go step by step to find the equation of the tangent line at the point (𝜋/4,𝜋/4) for the curve sin(𝑥−𝑦)=𝑥*cos(𝑦+𝜋/4).
Step 1: Find the derivative
Start by differentiating both sides of the equation with respect to x. Use the chain rule and product rule where necessary.
d/dx[sin(x - y)] = d/dx[x * cos(y + 𝜋/4)]
To differentiate sin(x - y), we need to use the chain rule. Let's differentiate each term separately:
d/dx[sin(x - y)] = cos(x - y) * (1 - dy/dx)
d/dx[x * cos(y + 𝜋/4)] = cos(y + 𝜋/4) * (1 - dy/dx) + x * (-sin(y + 𝜋/4)) * (dy/dx)
Step 2: Evaluate the derivatives at the given point
Substitute x = 𝜋/4 and y = 𝜋/4 into the expressions we found in step 1:
cos(𝜋/4 - 𝜋/4) * (1 - dy/dx) = 𝜋/4 * cos(𝜋/4 + 𝜋/4) * (1 - dy/dx) + 𝜋/4 * (-sin(𝜋/4 + 𝜋/4)) * (dy/dx)
Simplifying this equation:
cos(0) * (1 - dy/dx) = 𝜋/4 * cos(𝜋/2) * (1 - dy/dx) - 𝜋/4 * sin(𝜋/2) * (dy/dx)
1 - dy/dx = 𝜋/4 * 0 * (1 - dy/dx) - 𝜋/4 * 1 * (dy/dx)
1 - dy/dx = 0 - 𝜋/4 * dy/dx
Step 3: Solve for dy/dx
Rearrange the equation to solve for dy/dx:
1 = dy/dx * (𝜋/4 - 1)
dy/dx = 1 / (𝜋/4 - 1)
Step 4: Substitute the derivative into the point-slope form equation
Now that we have the value of dy/dx at the given point, substitute it into the point-slope form equation of a line:
y - 𝜋/4 = dy/dx * (x - 𝜋/4)
Substitute the values of dy/dx and the coordinates of the point (𝜋/4,𝜋/4):
y - 𝜋/4 = [1 / (𝜋/4 - 1)] * (x - 𝜋/4)
This equation represents the equation of the tangent line at the point (𝜋/4,𝜋/4).