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URL: https://nakafa.com/en/subjects/physics/vector/trigonometry-decomposition
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/physics/vector/trigonometry-decomposition/en.mdx

Learn how to resolve a slanted vector into horizontal and vertical components using sine and cosine.

---

## The Angle Determines the Components

When a vector forms an angle with an axis, its components can be found using trigonometry. The idea is simple: the slanted vector becomes the hypotenuse of a right triangle, while its components become the horizontal and vertical sides.

If angle $$\theta$$ is measured from the positive $$x$$ axis, the horizontal component is adjacent to the angle, while the vertical component is opposite the angle.

Visible text: If angle is measured from the positive axis, the horizontal component is adjacent to the angle, while the vertical component is opposite the angle.

Component: MathContainer
Children:

```math
F_x=F\cos\theta
```

```math
F_y=F\sin\theta
```

> Do not memorize where $$\sin$$ and $$\cos$$ go without checking the reference angle.

Visible text: > Do not memorize where and go without checking the reference angle.

## A Slanted Force on Coordinate Axes

Imagine a rope pulling a box with force $$\vec{F}$$ up and to the right. The pull has a horizontal part that pulls the box rightward and a vertical part that lifts slightly upward.

Visible text: Imagine a rope pulling a box with force up and to the right. The pull has a horizontal part that pulls the box rightward and a vertical part that lifts slightly upward.

Component: Vector3d
Props:
- title: Resolving a Slanted Force
- description: Force $$\vec{F}=50\text{ N}$$ forms a{" "}
$$37^\circ$$ angle above the horizontal, so its
components can be estimated from $$\cos37^\circ$$ and{" "}
$$\sin37^\circ$$.
  Visible text: Force forms a{" "}
 angle above the horizontal, so its
components can be estimated from and{" "}
.
- vectors: (() => {
const fx = [4, 0, 0];
const fy = [0, 3, 0];
const resultant = fx.map((component, index) => component + fy[index]);

return [
{
from: [0, 0, 0],
to: resultant,
color: getColor("VIOLET"),
label: "F",
labelAnchorX: "center",
labelProgress: 3 / 5,
labelOffset: [0.25, 0.5, 0.2],
},
{
from: [0, 0, 0],
to: fx,
color: getColor("TEAL"),
label: "Fx",
labelAnchorX: "center",
labelProgress: 1 / 2,
labelOffset: [0, -0.45, 0],
},
{
from: fx,
to: resultant,
color: getColor("AMBER"),
label: "Fy",
labelAnchorX: "center",
labelProgress: 1 / 2,
labelOffset: [0.55, 0, 0.2],
},
];
})()
- cameraPosition: [7, 5, 8]
- cameraTarget: [2, -0.8, 0]

The endpoint of $$F_x$$ followed by $$F_y$$ matches the endpoint of vector $$\vec{F}$$. The components are not extra forces. They are another way to describe the same force.

Visible text: The endpoint of followed by matches the endpoint of vector . The components are not extra forces. They are another way to describe the same force.

## Turning a Slanted Force into Components

A $$50\text{ N}$$ force acts $$37^\circ$$ above the horizontal. Use $$\cos37^\circ\approx0.8$$ and $$\sin37^\circ\approx0.6$$.

Visible text: A force acts above the horizontal. Use and .

Component: MathContainer
Children:

```math
\begin{aligned}
F_x&=F\cos\theta \\
&=50(0.8) \\
&=40\text{ N}
\end{aligned}
```

```math
\begin{aligned}
F_y&=F\sin\theta \\
&=50(0.6) \\
&=30\text{ N}
\end{aligned}
```

So the slanted $$50\text{ N}$$ force is equivalent to a $$40\text{ N}$$ pull to the right and a $$30\text{ N}$$ pull upward.

Visible text: So the slanted force is equivalent to a pull to the right and a pull upward.

The two components are not extra forces added to the situation. They are another way to describe the same force using horizontal and vertical directions.

## Angle from Vertical Axis

The formulas change when the reference angle changes. If angle $$\theta$$ is measured from the $$y$$ axis, the component adjacent to the angle is $$F_y$$.

Visible text: The formulas change when the reference angle changes. If angle is measured from the axis, the component adjacent to the angle is .

Component: MathContainer
Children:

```math
F_y=F\cos\theta
```

```math
F_x=F\sin\theta
```

Before using a formula, always ask: from which axis is the angle measured? This small check prevents a common error when resolving vectors.

## Component Signs Follow the Quadrant

The values of $$\sin$$ and $$\cos$$ give component sizes, but positive or negative signs come from the vector direction on the coordinate axes.

Visible text: The values of and give component sizes, but positive or negative signs come from the vector direction on the coordinate axes.

| Vector direction | Sign of $$F_x$$ | Sign of $$F_y$$ |
| :--------------- | :------------------------------- | :------------------------------- |
| up and right | positive | positive |
| up and left | negative | positive |
| down and left | negative | negative |
| down and right | positive | negative |

Visible text: | Vector direction | Sign of | Sign of |
| :--------------- | :------------------------------- | :------------------------------- |
| up and right | positive | positive |
| up and left | negative | positive |
| down and left | negative | negative |
| down and right | positive | negative |

If a force points up and to the left, its horizontal component is negative because it opposes the positive $$x$$ axis, while its vertical component remains positive.

Visible text: If a force points up and to the left, its horizontal component is negative because it opposes the positive axis, while its vertical component remains positive.