# Nakafa Framework: LLM

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

Output docs content for large language models.

---

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

export const metadata = {
  title: "Trigonometric Comparison: Tan θ",
  description: "Calculate tangent ratios using opposite and adjacent sides. Solve height problems, slopes, and angles with step-by-step examples and practice exercises.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/14/2025",
  subject: "Trigonometry",
};

## What is Tangent in Trigonometric Ratios?

When studying trigonometry, we encounter several types of ratios. One of the most fundamental is the tangent ratio, often abbreviated as tan.

### Understanding Tangent (tan)

Tangent is the ratio between the length of the opposite side (the side opposite to the known angle) and the adjacent side (the side adjacent to the angle) in a right-angled triangle.

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

<div className="mt-4">
  <Triangle
    title="Visualization of Tangent in a Triangle"
    description="Tangent is the ratio of the opposite side to the adjacent side."
    angle={30}
    labels={{
      opposite: "opposite side",
      adjacent: "adjacent side",
      hypotenuse: "hypotenuse",
    }}
  />
</div>

In a right-angled triangle, we can observe that:

- The opposite side is the side facing the angle θ
- The adjacent side is the side next to the angle θ (other than the hypotenuse)
- Tangent is calculated by dividing the length of the opposite side by the adjacent side

### Examples of Tangent Values

Tangent values for specific angles can be calculated precisely. For example:

- <InlineMath math="\tan 30° = 0.57 \text{ or } \frac{1}{\sqrt{3}}" />
- <InlineMath math="\tan 45° = 1" />
- <InlineMath math="\tan 60° = 1.73 \text{ or } \sqrt{3}" />

These tangent values can be obtained by calculating the ratio of sides in triangles with these angles.

## Calculating Tangent Values

### Example of Calculating Tangent

For instance, if we have a right-angled triangle with an angle of <InlineMath math="30°" />, an opposite side of 15 cm, and an adjacent side of 26 cm.

<Triangle
  title={
    <>
      Example Triangle with <InlineMath math="30°" /> Angle
    </>
  }
  description="Ratio of opposite side (15 cm) to adjacent side (26 cm)."
  angle={30}
  labels={{
    opposite: "15 cm",
    adjacent: "26 cm",
    hypotenuse: "hypotenuse",
  }}
/>

The tangent value of this angle is:

<BlockMath math="\tan 30° = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{15 \text{ cm}}{26 \text{ cm}} = 0.57" />

### Applications in Everyday Life

Tangent is very useful in everyday life, especially for:

1. Calculating the height of objects (such as buildings, trees) from a certain distance
2. Determining the slope (gradient) of roads or stairs
3. In architecture for calculating roof angles
4. Navigation and direction determination

## Practice Exercise

Consider the following right-angled triangle with an angle of <InlineMath math="70°" />, an opposite side of 15 cm, and an adjacent side of 7 cm.

<Triangle
  title={
    <>
      Triangle with <InlineMath math="70°" /> Angle
    </>
  }
  description="Triangle with opposite side 15 cm and adjacent side 7 cm."
  angle={70}
  labels={{
    opposite: "15 cm",
    adjacent: "7 cm",
    hypotenuse: "hypotenuse",
  }}
/>

Can you find the tangent ratio value of <InlineMath math="\tan 70°" />? Explain why!

### Answer Key

Yes, we can find the value of <InlineMath math="\tan 70°" /> by dividing the opposite side by the adjacent side:

<BlockMath math="\tan 70° = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{15 \text{ cm}}{7 \text{ cm}} = 2.14" />

Therefore, the value of <InlineMath math="\tan 70° = 2.14" />.
