A Level Maths: Introduction To Resolving Forces

Do you ever stare at one of those slope diagrams and think, why are there so many arrows?
Same. Everyone does it for the first time.
All that’s really happening is one slanted force being split into two directions. Across. Up. That’s it.
Once you see that, half of Mechanics stops looking like black magic.

🔙 Previous topic:

“Review projectile examples before before continuing with resolving forces.”

🧭 Why this keeps showing up

Right, so it doesn’t matter if it’s Year 12 or 13, every paper sneaks this in somewhere.
On the flat, on an incline, or tied to a string — at some point you’ll be breaking a force apart.
The maths? Old friend from GCSE: SOH CAH TOA.
Actually, hold on, that’s the good bit — you already know the trig; you’re just using it on forces now.

⚙️ Resultant forces — the idea

A force changes motion. Measured in newtons.
Weight’s the classic one: weight = mass × g, with g about 9.8 m/s².

If a book just sits on a table, gravity drags it down and the table pushes back up with an equal normal reaction R.
They cancel. No movement.
So far, nothing fancy.

Now, when you’ve got two or more forces pulling on the same object, you can replace them with one single resultant.
That one’s easier to work with — same effect, fewer arrows.

📏 Quick test

Two pulls: 50 N and 10 N, both to the right.
Together they act as 60 N.
If one was left instead, you’d subtract.
I sometimes call this “tug-of-war maths”. Stronger side wins; the difference is your resultant.

🧠 What “resolving” really means

When a force sits at an angle, we split it into two bits at right angles — horizontal and vertical.
That’s resolving. Literally breaking one arrow into two tidy ones.

Formula time:
Horizontal part = F cos θ
Vertical part = F sin θ

Right, let’s try one before the theory drifts away.

📘 Small worked example

Force of 10 N acting 30° above the horizontal.

Across: 10 cos 30° = 8.66 N
Up: 10 sin 30° = 5 N

So those two numbers together give the same effect as the single 10 N at an angle.
Nice, clean triangle.
If you check with Pythagoras: 8.66² + 5² = 10². Works perfectly.

⚙️ Going the other way

Now say you know the components and want the single force.
Reverse the process: Resultant = √(Fx² + Fy²).
Angle = tan⁻¹(Fy / Fx).

Plug the same numbers back in — √(8.66² + 5²) = 10 N, angle 30°.
So you’ve just gone full circle. Split and re-combine.

📘 Bit more crowded — three forces

Okay, deep breath.
This one’s where sign errors start creeping in.
Pick your positive direction first — seriously, do it before writing any numbers.
Let’s say right and upward are positive.

Now resolve:
Horizontal → 4 cos 30 + 3 cos 10 − 5 sin 20 = 4.708 N
Vertical → 4 sin 30 − 3 sin 10 − 5 cos 20 = −3.219 N

Resultant = √(4.708² + 3.219²) ≈ 5.7 N.
Angle = tan⁻¹(3.219 / 4.708) ≈ 34° below horizontal.

If your sign flips, don’t panic. Just means your first guess for direction was opposite — you can still get full marks if you’re consistent.

⚙️ Vectors show up

Sometimes questions ditch the words and jump straight to ai + bj.
No drama. i is across, j is up.
Exactly the same thing.

Example: 4 N at 30°.
Across = 4 cos 30 = 3.46 N.
Up = 4 sin 30 = 2 N.
Write it as 3.46i + 2j N.
Done.

📏 Adding a few at once

Let’s pile three together.
F₁ = 2i + 3j, F₂ = i − 2j, F₃ = 2i − 6j (all newtons).

Add them: (2 + 1 + 2)i + (3 − 2 − 6)j = 5i − 5j.
Magnitude = √(5² + 5²) = 7.07 N.
Direction 45° below horizontal.

Easy pattern once you’ve drawn it once.
Still just a right-angled triangle.

❗ Equilibrium — when nothing moves

If all those forces balance so the resultant is zero, the body’s in equilibrium.
Either it’s sitting still or gliding at constant speed.
Same rule.

Let’s check one quickly.
Forces: 6 N at 30°, a force P at angle θ, and a 9 N horizontal pull.
Equilibrium means everything cancels.

Horizontally: 6 cos 30 + P cos θ − 9 = 0 → P cos θ = 3.80
Vertically: 6 sin 30 − P sin θ = 0 → P sin θ = 3

Now tan θ = 3 / 3.80 = 0.79 → θ ≈ 38°.
Then P = 3 / sin 38 = 4.8 N.
Looks neat. Balanced both ways.

🧠 Mistakes everyone makes

Forgetting to decide which way is positive.
Mixing up sin and cos with the given angle.
Missing a minus sign halfway.
Leaving out N.
Working in radians by accident.

Write “→ right is positive” at the top of every question. It keeps your head clear.

🧠 Teacher aside

Every year someone asks, “Do I really have to draw the triangle?”
Yes. Yes, you do.
I once had an entire class redo a question because half had their angle drawn on the wrong side of the line.
Five minutes later they all got it right.
That’s how powerful the picture is.

So don’t skip the sketch — even the rough one on scrap paper.

📘 Real-world tie-in

Hold a heavy bag at an angle and feel the pull.
Part of that’s horizontal, part vertical — that’s resolving forces with your shoulder.
Engineers do the same thing for bridges, cranes, scaffolds, car suspensions.
Same triangles, bigger numbers.

🚀 Next steps

If this still feels a bit slippery, our A Level Booster Course walks through these setups slowly — slopes, friction, strings — one clean diagram at a time.
🚀 Same conversational tone, same plain-text maths, all exam-board aligned.

Anyway, try making up one problem yourself tonight.
Pick a force, choose an angle, split it, re-combine it.
Talk it out loud as you do it — you’ll be surprised how much you remember next week.

✅ Quick recap table

Concept	Formula / Rule	Tip
Horizontal comp.	F cos θ	Adjacent side
Vertical comp.	F sin θ	Opposite side
Resultant	√(Fx² + Fy²)	Pythagoras again
Direction	tan⁻¹(Fy / Fx)	Watch the sign
Equilibrium	ΣF = 0	Forces cancel
Vector form	ai + bj	i → x, j → y

About the Author

S. Mahandru is the Head of Mathematics at Exam.tips, specialising in A Level and GCSE Mathematics education. With over a decade of classroom and online teaching experience, he has helped thousands of students achieve top results through clear explanations, practical examples, and applied learning strategies.

Updated: November 2025

🧭 Next topic:

“Next, see how all this applies in real engineering contexts.”

A Level Maths: Introduction To Resolving Forces