# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/10/mathematics/trigonometry/trigonometric-comparison-tan-usage
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/trigonometry/trigonometric-comparison-tan-usage/en.mdx

Output docs content for large language models.

---

import { Triangle } from "@repo/design-system/components/contents/triangle";

export const metadata = {
  title: "Applications of Tan θ Trigonometric Comparison",
  description: "Apply tangent to measure building heights using shadows. Master similar triangles and elevation angles for real-world distance and height calculations.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/14/2025",
  subject: "Trigonometry",
};

## What is Tangent?

In a right triangle, the tangent of angle θ 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.

<Triangle
  title={
    <>
      Visualization of Tangent (<InlineMath math="\tan \theta" />)
    </>
  }
  description="Slide the slider to see how tangent 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 θ, then:

<BlockMath math="\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}" />

The value of tangent changes according to the angle. For example, <InlineMath math="\tan 30° = 0.57 \text{ or } \frac{1}{\sqrt{3}}" />.

## 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 (θ)
3. Use the tan θ formula to calculate the object's height (opposite side)

<div className="mt-4">
  <BlockMath math="\text{object height} = \text{shadow length} \times \tan \theta" />
</div>

<div className="mt-4">
  <Triangle
    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: "",
    }}
  />
</div>

## 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:

<BlockMath 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 cm), their shadow length (200 cm), and the teenager's height (148 cm), we can calculate the teenager's shadow length:

<div className="flex flex-col gap-4">
  <BlockMath math="\frac{114 \text{ cm}}{200 \text{ cm}} = \frac{148 \text{ cm}}{x \text{ cm}}" />
  <BlockMath 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}" />
</div>

### Using the Tangent Formula

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

<BlockMath math="\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}" />

For example, if we have a sun elevation angle of 30° (<InlineMath math="\tan 30° = 0.57" />) and want to calculate the shadow length of a teenager with height 148 cm:

<div className="flex flex-col gap-4">
  <BlockMath math="\tan \theta = \frac{\text{teenager's height}}{\text{teenager's shadow length}}" />
  <BlockMath math="\tan 30° = \frac{148 \text{ cm}}{x \text{ cm}}" />
  <BlockMath math="0.57 = \frac{148 \text{ cm}}{x \text{ cm}}" />
  <BlockMath math="x \text{ cm} = \frac{148 \text{ cm}}{0.57} = 259.65 \text{ cm}" />
</div>

<div className="mt-4">
  <Triangle
    title="30 Degree Angle with Tangent 0.57"
    description={
      <>
        Visualization of a triangle with a <InlineMath math="30°" /> angle as in
        the calculation example.
      </>
    }
    angle={30}
    labels={{
      opposite: "Teenager's Height",
      adjacent: "Teenager's Shadow Length",
      hypotenuse: "",
    }}
  />
</div>
