We are doing the force table lab in class and now we are calculating the f equilibran and the equilibrant angle. I had no time graphing a solution and came up with the right answers by measuring, but we are also suppose to do it purely mathematically and I do not know how to. I assume I will have to find vector componments, but I cannot get a right triangle in the first place. Please explain step by step.

Here are my problems:
100g 30* and 200g 220*

100g 30* and 200g 120* and 150g 330*

To calculate the force required to balance a force table experiment mathematically, you need to use vector components and trigonometry. Let's walk through the process step by step using your problem examples.

1. Convert the given angles to their respective vector components:
For the first example:
- A force of 100g at 30° will have x-component = 100g * cos(30°) and y-component = 100g * sin(30°).
- A force of 200g at 220° will have x-component = 200g * cos(220°) and y-component = 200g * sin(220°).

2. Calculate the sum of the x-components and y-components separately:
- For the first example:
- X-component sum: x1 + x2 = (100g * cos(30°)) + (200g * cos(220°))
- Y-component sum: y1 + y2 = (100g * sin(30°)) + (200g * sin(220°))

3. Determine the magnitude of the resultant force:
- Use the Pythagorean theorem to calculate the magnitude of the resultant force:
- Magnitude = sqrt((x-component sum)^2 + (y-component sum)^2)
- For the first example: Magnitude = sqrt((x1 + x2)^2 + (y1 + y2)^2)

4. Calculate the angle of the equilibrant force:
- Use the inverse tangent function (tan^(-1)) to find the angle of the equilibrant force relative to the positive x-axis:
- Angle = tan^(-1)(y-component sum / x-component sum)
- For the first example: Angle = tan^(-1)((y1 + y2) / (x1 + x2))

Now let's solve the first example step by step:

Given forces: 100g at 30° and 200g at 220°

1. Convert the angles to vector components:
- x1 = 100g * cos(30°)
- y1 = 100g * sin(30°)
- x2 = 200g * cos(220°)
- y2 = 200g * sin(220°)

2. Calculate the sum of the x-components and y-components:
- x-component sum = x1 + x2
- y-component sum = y1 + y2

3. Determine the magnitude of the resultant force:
- Magnitude = sqrt((x1 + x2)^2 + (y1 + y2)^2)

4. Calculate the angle of the equilibrant force:
- Angle = tan^(-1)((y1 + y2) / (x1 + x2))

Repeat the same process for the second example with three given forces - 100g at 30°, 200g at 120°, and 150g at 330°.

Remember to convert the angles to their respective vector components and then perform the necessary calculations.