Example (1): Resolving a force of 10 N at 30°

• Find the horizontal and vertical components of a force of 10 N that makes an angle of 30° with the positive x-axis.
• Solution:

1. F_x = 10 × cos 30° = 10 × √3 ÷ 2 = 5√3 ≈ 8.66 N.
2. F_y = 10 × sin 30° = 10 × 1 ÷ 2 = 5 N.
3. 8.66² + 5² = 75 + 25 = 100 = 10², which is correct.

Example (2): Resolving a force of 20 N at 60°

• A force of 20 N makes an angle of 60° with the x-axis. Find its components.
• Solution:

1. F_x = 20 × cos 60° = 20 × 1 ÷ 2 = 10 N.
2. F_y = 20 × sin 60° = 20 × √3 ÷ 2 = 10√3 ≈ 17.32 N.
3. Notice that the vertical component is larger than the horizontal one because the angle is greater than 45°.

Example (3): A force making an angle of 45°

1. A person pulls a box with a rope whose tension is 12 N.
2. The rope makes an angle of 45° with the horizontal.
3. Find the horizontal component of the tension.
4. Solution:

1. Horizontal component = 12 × cos 45° = 12 × √2 ÷ 2 = 6√2.
2. 6√2 ≈ 8.49 N.
3. In this case the vertical component is also equal to 6√2 N, because sin 45° = cos 45°.

Example (4): Finding a force from its components

• A force has a horizontal component of 6 N to the right and a vertical component of 8 N upwards.
• Find the magnitude and the direction of the force.
• Solution:

1. F = √(6² + 8²) = √100 = 10 N.
2. tan θ = 8 ÷ 6 = 4 ÷ 3, so θ ≈ 53.13°.
3. The force of 10 N makes an angle of about 53.13° with the positive x-axis.