# Nakafa Framework: LLM

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

Output docs content for large language models.

---

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

export const metadata = {
  title: "Special Angles in Trigonometric Comparisons",
  description: "Memorize exact trigonometric values for 30°, 45°, and 60° angles. Get precise sin, cos, tan ratios without calculators using unit circle visualization.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/14/2025",
  subject: "Trigonometry",
};

## What are Special Angles?

Special angles in trigonometric ratios are angles whose ratio values can be determined exactly (precise values) without using a calculator. These values are very important to remember and will be frequently used in various mathematical and physics calculations.

The most common special angles are <InlineMath math="0°" />, <InlineMath math="30°" />, <InlineMath math="45°" />, <InlineMath math="60°" />, and <InlineMath math="90°" />.

<UnitCircle
  title="Visualization of Special Angles on the Unit Circle"
  description={
    <>
      Move the slider to angles <InlineMath math="0°" />,{" "}
      <InlineMath math="30°" />, <InlineMath math="45°" />,{" "}
      <InlineMath math="60°" />, or <InlineMath math="90°" /> to see the
      positions of special angles.
    </>
  }
  angle={45}
/>

## Exact Values of Trigonometric Ratios for Special Angles

Below is a table of exact values for trigonometric ratios at special angles:

| Ratio                             | <InlineMath math="0°" /> | <InlineMath math="30°" />                                     | <InlineMath math="45°" />                                     | <InlineMath math="60°" />                | <InlineMath math="90°" /> |
| --------------------------------- | ------------------------ | ------------------------------------------------------------- | ------------------------------------------------------------- | ---------------------------------------- | ------------------------- |
| <InlineMath math="\sin \theta" /> | <InlineMath math="0" />  | <InlineMath math="\frac{1}{2}" />                             | <InlineMath math="\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}" /> | <InlineMath math="\frac{\sqrt{3}}{2}" /> | <InlineMath math="1" />   |
| <InlineMath math="\cos \theta" /> | <InlineMath math="1" />  | <InlineMath math="\frac{\sqrt{3}}{2}" />                      | <InlineMath math="\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}" /> | <InlineMath math="\frac{1}{2}" />        | <InlineMath math="0" />   |
| <InlineMath math="\tan \theta" /> | <InlineMath math="0" />  | <InlineMath math="\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}" /> | <InlineMath math="1" />                                       | <InlineMath math="\sqrt{3}" />           | Undefined                 |

## Origin of Exact Values

The exact values for special angles are derived from:

1. **Angles <InlineMath math="30°" /> and <InlineMath math="60°" />**: Obtained from a right triangle with side ratio <InlineMath math="1 : 2 : \sqrt{3}" />.
2. **Angle <InlineMath math="45°" />**: Obtained from an isosceles right triangle with side ratio <InlineMath math="1 : 1 : \sqrt{2}" />.