Find the maximum or minimum values of sin x+sin y+sin(x+y)

z = sinx + siny + sin(x+y)

to find the maxima, we need

∂z/∂x = 0 and ∂z/∂y = 0

∂z/∂x = cosx + cos(x+y)
∂z/∂y = cosy + cos(x+y)

∂z/dx = 0 when x+y = pi-x
A handy place for this is at (pi/3,pi/3)

z(pi/3,pi/3) = √3/2 + √3/2 + √3/2 = 3√3/2 = 2.598

To find the maximum or minimum values of the expression sin x + sin y + sin(x + y), we can use some properties of trigonometric functions. Let's analyze the function step by step:

Step 1: Restrictions on x and y
Since we are dealing with trigonometric functions, the values of x and y can range from -∞ to +∞.

Step 2: Rewrite the expression using trigonometric identities
sin(x + y) can be rewritten as sin x cos y + cos x sin y using the angle addition formula for sine. Therefore, the given expression becomes:

sin x + sin y + sin x cos y + cos x sin y

Step 3: Arrange the terms
Rearranging the terms, we get:

sin x + sin x cos y + sin y + cos x sin y

Step 4: Factor out sin x and sin y
Factoring out sin x and sin y, we have:

sin x(1 + cos y) + sin y(1 + cos x)

Step 5: Analyzing the terms
The maximum value of sin x is 1, and the maximum value of sin y is also 1. The maximum value of cos y occurs when y = 0, which is also equal to 1. Similarly, the maximum value of cos x occurs when x = 0.

Step 6: Final Analysis
To find the maximum value of the given expression, we substitute the maximum values of sin x, sin y, cos y, and cos x into the expression:

sin x(1 + cos y) + sin y(1 + cos x)
= 1(1 + 1) + 1(1 + 1)
= 1(2) + 1(2)
= 2 + 2
= 4

Therefore, the maximum value of sin x + sin y + sin(x + y) is 4.

There is no minimum value for the given expression since the values of sin x, sin y, cos y, and cos x are bounded between -1 and 1, and there are no restrictions on x and y.

To find the maximum or minimum values of the function sin x + sin y + sin(x + y), we can use the properties of trigonometric functions and calculus. First, let's consider the range of values that sin x, sin y, and sin(x + y) can take.

For sine function, the range is from -1 to 1. So sin x, sin y, and sin(x + y) individually can only take values between -1 and 1.

To find the maximum or minimum values of the given function, we need to find the critical points. These are the points where the derivative of the function is either zero or undefined.

Taking the partial derivative of the function with respect to x and y, we have:

∂/∂x (sin x + sin y + sin(x + y)) = cos x + cos(x + y) = 0,
∂/∂y (sin x + sin y + sin(x + y)) = cos y + cos(x + y) = 0.

Simplifying these equations gives:
cos x + cos(x + y) = 0,
cos y + cos(x + y) = 0.

Now, we can solve these equations to find the values of x and y that satisfy them. However, since the equation involves a combination of sine and cosine functions, it is not easy to find exact solutions.

Instead, we can make use of calculators or numerical methods to approximate the values of x and y that satisfy the equations.

Once we find the critical points (x, y), we can substitute these values back into the original function to determine whether they correspond to maximum or minimum values.

To summarize:
1. Calculate the partial derivatives of the given function with respect to x and y.
2. Set the derivatives equal to zero and solve for x and y.
3. Substitute the critical points into the original function to find maximum or minimum values.
4. Use calculators or numerical methods if exact solutions are not feasible.

Note: It's important to keep in mind that there might be multiple maximum or minimum values depending on the range of x and y.