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URL: https://nakafa.com/en/subjects/mathematics/trigonometry/trigonometric-comparison-tan-usage
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Apply tangent to measure building heights using shadows. Learn similar triangles and elevation angles for real-world distance and height calculations.

---

## What is Tangent?

In a right triangle, the tangent of angle $$\theta$$ is the ratio between the length of the opposite side and the adjacent side. This is very different from sine, which compares the opposite side to the hypotenuse, or cosine, which compares the adjacent side to the hypotenuse.

Visible text: In a right triangle, the tangent of angle is the ratio between the length of the opposite side and the adjacent side. This is very different from sine, which compares the opposite side to the hypotenuse, or cosine, which compares the adjacent side to the hypotenuse.

Component: Triangle
Props:
- title: Visualization of Tangent ($$\tan \theta$$)
  Visible text: Visualization of Tangent ()
- description: Slide the slider to see how $$\tan$$ changes as the angle changes.
  Visible text: Slide the slider to see how changes as the angle changes.
- angle: 30
- labels: {
opposite: "Opposite Side",
adjacent: "Adjacent Side",
hypotenuse: "Hypotenuse",
}

For example, if we have a right triangle with angle $$\theta$$, then:

Visible text: For example, if we have a right triangle with angle , then:

```math
\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}
```

The value of tangent changes according to the angle. For example, $$\tan 30^\circ = 0.57 \text{ or} \frac{1}{\sqrt{3}}$$.

Visible text: The value of tangent changes according to the angle. For example, .

## Applications of Tangent in Daily Life

The tangent trigonometric ratio is very useful for solving various real-life problems, especially when we want to calculate:

1. Height of objects that are difficult to measure directly
2. Distance between two points that cannot be accessed
3. Length of an object's shadow
4. Slope of a surface

### Measuring Height through Shadows

Imagine we want to measure the height of a tree, building, or other tall object. We can use the tangent ratio with the following steps:

1. Measure the length of the object's shadow (adjacent side)
2. Measure or know the sun's angle of elevation $$\theta$$.
3. Use the $$\tan \theta$$ formula to calculate the object's height (opposite side).

Visible text: 1. Measure the length of the object's shadow (adjacent side)
2. Measure or know the sun's angle of elevation .
3. Use the formula to calculate the object's height (opposite side).

Component: MathContainer
Children:

```math
\text{object height} = \text{shadow length} \times \tan \theta
```

Component: ContentBlock
Children:
Component: Triangle
Props:
- title: Application of Tangent in Measurement
- description: Example of a triangle formed when measuring height with shadows.
- angle: 30
- labels: {
opposite: "Object Height",
adjacent: "Shadow Length",
hypotenuse: "",
}

## Methods for Calculating with Tangent

There are two ways we can use to solve problems using tangent:

### Similar Triangle Comparison

We can use the principle of similar triangles to solve problems. If we have two triangles with the same shape (similar), then the ratio of their sides will be the same.

For example, if we have shadows from three objects of different heights (child, teenager, and adult), we can create the equation:

```math
\frac{\text{child's height}}{\text{child's shadow length}} = \frac{\text{teenager's height}}{\text{teenager's shadow length}} = \frac{\text{adult's height}}{\text{adult's shadow length}}
```

With this equation, if we know the child's height (e.g., $$114 \text{ cm}$$), their shadow length ($$200 \text{ cm}$$), and the teenager's height ($$148 \text{ cm}$$), we can calculate the teenager's shadow length:

Visible text: With this equation, if we know the child's height (e.g., ), their shadow length (), and the teenager's height (), we can calculate the teenager's shadow length:

Component: MathContainer
Children:

```math
\frac{114 \text{ cm}}{200 \text{ cm}} = \frac{148 \text{ cm}}{x \text{ cm}}
```

```math
x \text{ cm} = \frac{148 \text{ cm} \times 200 \text{ cm}}{114 \text{ cm}} = \frac{29{,}600 \text{ cm}}{114 \text{ cm}} = 259.65 \text{ cm}
```

### Using the Tangent Formula

Another more direct way is to use the tangent trigonometric ratio. We know that:

```math
\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}
```

For example, if we have a sun elevation angle of $$30^\circ$$ ($$\tan 30^\circ = 0.57$$) and want to calculate the shadow length of a teenager with height $$148 \text{ cm}$$:

Visible text: For example, if we have a sun elevation angle of () and want to calculate the shadow length of a teenager with height :

Component: MathContainer
Children:

```math
\tan \theta = \frac{\text{teenager's height}}{\text{teenager's shadow length}}
```

```math
\tan 30^\circ = \frac{148 \text{ cm}}{x \text{ cm}}
```

```math
0.57 = \frac{148 \text{ cm}}{x \text{ cm}}
```

```math
x \text{ cm} = \frac{148 \text{ cm}}{0.57} = 259.65 \text{ cm}
```

Component: ContentBlock
Children:
Component: Triangle
Props:
- title: Angle $$30^\circ$$ with Tangent $$0.57$$
  Visible text: Angle with Tangent
- description: Visualization of a triangle with a $$30^\circ$$ angle as in
the calculation example.
  Visible text: Visualization of a triangle with a angle as in
the calculation example.
- angle: 30
- labels: {
opposite: "Teenager's Height",
adjacent: "Teenager's Shadow Length",
hypotenuse: "",
}