Resolving a force splits it into two perpendicular parts. The part next to the given angle uses cosine, and the part opposite the angle uses sine, so the triangle decides which to use.
This lesson is part of SPM Physics force and motion II. It is the base for combining forces and finding a resultant.
What is the triangle-first method?
Use these steps.
- Draw the force as the hypotenuse of a right-angled triangle.
- Mark the given angle and its reference line, such as the horizontal.
- Label the side next to the angle with cos and the side opposite with sin.
- Write each component as force × cos θ or force × sin θ.
The method avoids relying on memory of “horizontal uses cos”. On a slope, the rule changes, and the triangle shows why.
Worked example 1: a pulled box
A 50 N force pulls a box at 30° above the horizontal. Find the horizontal and vertical components.
The horizontal side touches the 30° angle, so Fx = 50 cos 30° = 50 × 0.866 = 43.3 N. The vertical side is opposite the angle, so Fy = 50 sin 30° = 50 × 0.5 = 25 N.
Check: √(43.3² + 25²) = √(1875 + 625) = √2500 = 50 N, which matches the original force.
Worked example 2: a block on a slope
A 6.0 kg block rests on a slope of 30°. Take g = 10 m s⁻², so the weight is 60 N, acting vertically downward.
The weight makes a 30° angle with the line perpendicular to the slope. The component along the slope is the side opposite that angle, so it is 60 sin 30° = 30 N. The component perpendicular to the slope is next to the angle, so it is 60 cos 30° = 52 N, correct to two significant figures.
The component along the slope tries to slide the block down, and the component perpendicular to the slope presses it into the surface.
The mistake to avoid
The common mistake is to use the same function for a slope and a flat floor. Compare the two cases.
| Situation | Angle measured from | Component along the surface |
|---|---|---|
| Force pulled above the horizontal | the horizontal | F cos θ |
| Weight on a slope of angle θ | the perpendicular to the slope | W sin θ |
The slope angle is the same as the angle between the weight and the perpendicular to the slope. Draw the small triangle to confirm it. The units and significant figure checker helps you keep the answers in newtons.
Check yourself
A force of 80 N acts at 37° above the horizontal. Find its horizontal and vertical components. Use sin 37° = 0.60 and cos 37° = 0.80.
Answer
Horizontal: 80 cos 37° = 80 × 0.80 = 64 N. Vertical: 80 sin 37° = 80 × 0.60 = 48 N.
Check: √(64² + 48²) = √(4096 + 2304) = √6400 = 80 N.
What to study next
Continue with combining forces and finding a resultant, which reverses this process. Then apply both ideas in applying equilibrium conditions.
If you want a teacher to watch you choose sine or cosine, see online one-to-one Physics tuition.