Lessons · Engineering · vectors and their components
A force has a direction, so split it into two you can add
Any force can be replaced by a horizontal part F cos θ and a vertical part F sin θ, and parts along the same axis simply add.
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
What it is for
A guy wire pulls on a pole at an angle. The concrete cares only about the downward part and the pole cares only about the sideways part. Until the pull is split, neither question can be answered.
How to think about it
Draw the force, draw the angle it makes with the horizontal, and draw the right triangle. The side next to the angle takes the cosine; the side across from it takes the sine. Check that the two parts recombine to the original.
Worked example
F = 200 N at 60° above horizontalOne force, one angle from the horizontal axis.
F_x = F cos θ = 200 × cos 60° = 200 × 0.5 = 100 NThe horizontal part.
F_y = F sin θ = 200 × sin 60° = 200 × 0.866 = 173.2 NThe vertical part. Steeper angle, bigger vertical share.
Check: sqrt(100² + 173.2²) = sqrt(10,000 + 30,000) = 200 NThe two parts recombine to the force you started with. Always do this line.
Your turn
A 400 N pull at 30° above horizontal. Write the horizontal part.
F_x = 400 × cos = 346.4 N
Solve one, graded on the server
The trap
Swapping sine and cosine. Draw the triangle every time until you no longer need to; the part next to the angle is the cosine, and the part that would vanish if the angle went to zero is the sine.