How can we eliminate the parameter from the equation x sin(1/2) + y cos(1/2) to find the Cartesian equation of the curve?

To eliminate the parameter in the equation x sin(1/2) + y cos(1/2) = p, we first need to understand the components involved:

1. Identifying Parameters: Here, x, y, and p are the variables, while the equation includes trigonometric functions of a constant (1/2).
2. Rearranging the Equation: We can rearrange the equation to isolate p:

          p = x sin(1/2) + y cos(1/2)

3. Substituting the Values: Since we have two separate components (one with x and the other with y), we will express these in a Cartesian form. We know that sin and cos can be related to a circle:

Let’s denote:

  a = sin(1/2) 

  b = cos(1/2) 

Then the equation becomes:

          p = ax + by 

4. Expressing in Cartesian Form: Now if we set other relationships or conditions based on certain values of p, we can transform this:

  a^2 + b^2 = 1

This follows from the basic identity of the sine and cosine functions.

5. Finalizing the Cartesian Equation: To eliminate the parameter effectively, consider substituting coefficients into a Cartesian coordinate system. By using various values and relationships, we ultimately arrive at:

The Cartesian equation can be expressed as:

  x^2/a^2 + y^2/b^2 = 1

This represents an ellipse or circle, derived from the original trigonometric parameters given in the question.

In summary, we have identified the parameters, rearranged the equation, substituted known relationships, and finally emerged with a clean Cartesian equation.