Have sin cos cos sin sin for the last term

Solved Step by Step With Explanation- Tangent slope at (7π,3π/2): 1

Questions

at the point (7π,3π/2).

Answer

We want to find dy/dx, which represents the slope of the tangent line at the point (7π, 3π/2). Here are the steps:

Step 1: Differentiate both sides of the equation with respect to x.

For the second term, 4cos(y), we need to use the chain rule because y is a function of x. The derivative of cos(y) with respect to y is -sin(y), and the derivative of y with respect to x is dy/dx. So, we have:

d/dx [4cos(y)] = -4sin(y) * dy/dx

5cos(x) - 4sin(y)dy/dx - (-3cos(x)cos(y) + 3sin(x)sin(y)) + 1 = 0

Step 3: Simplify the equation.

5cos(7π) + 3cos(7π)cos(3π/2) - 3sin(7π)sin(3π/2) - 4sin(3π/2)dy/dx + 1 = 0

Now, evaluate the trigonometric functions:

Plug these values into the equation:

5(-1) + 3(0)(-1) - 3(0)(-1) - 4(-1)dy/dx + 1 = 0

Now, solve for dy/dx:

4dy/dx = 4