# Nakafa Framework: LLM

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

Output docs content for large language models.

---

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

export const metadata = {
  title: "Trigonometric Comparison: Sin θ and Cos θ",
  description: "Master sine and cosine ratios with real-world pyramid examples. Compare trigonometric functions, solve practical problems, and understand their applications.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/14/2025",
  subject: "Trigonometry",
};

## What is Sine Ratio (sin θ)?

Sine of an angle θ in a right triangle is the ratio between the length of the opposite side and the hypotenuse.

<BlockMath math="\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}" />

<div className="mt-4">
  <Triangle
    title={
      <>
        Visualization of Sine (<InlineMath math="\sin \theta" />)
      </>
    }
    description="Move the slider to see how the sine changes as the angle changes."
    angle={30}
    labels={{
      opposite: "Opposite Side",
      adjacent: "Adjacent Side",
      hypotenuse: "Hypotenuse",
    }}
  />
</div>

### What is Cosine Ratio (cos θ)?

Cosine of an angle θ in a right triangle is the ratio between the length of the adjacent side and the hypotenuse.

<BlockMath math="\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}" />

<div className="mt-4">
  <Triangle
    title={
      <>
        Visualization of Cosine (<InlineMath math="\cos \theta" />)
      </>
    }
    description="Move the slider to see how the cosine changes as the angle changes."
    angle={60}
    labels={{
      opposite: "Opposite Side",
      adjacent: "Adjacent Side",
      hypotenuse: "Hypotenuse",
    }}
  />
</div>

## Sine and Cosine Values for Common Angles

Here are some sine and cosine values for commonly used angles:

| Angle                     | Sine Value (sin θ)                       | Decimal Value              | Cosine Value (cos θ)                     | Decimal Value              |
| ------------------------- | ---------------------------------------- | -------------------------- | ---------------------------------------- | -------------------------- |
| <InlineMath math="0°" />  | <InlineMath math="0" />                  | <InlineMath math="0" />    | <InlineMath math="1" />                  | <InlineMath math="1" />    |
| <InlineMath math="30°" /> | <InlineMath math="\frac{1}{2}" />        | <InlineMath math="0.5" />  | <InlineMath math="\frac{\sqrt{3}}{2}" /> | <InlineMath math="0.87" /> |
| <InlineMath math="45°" /> | <InlineMath math="\frac{\sqrt{2}}{2}" /> | <InlineMath math="0.71" /> | <InlineMath math="\frac{\sqrt{2}}{2}" /> | <InlineMath math="0.71" /> |
| <InlineMath math="60°" /> | <InlineMath math="\frac{\sqrt{3}}{2}" /> | <InlineMath math="0.87" /> | <InlineMath math="\frac{1}{2}" />        | <InlineMath math="0.5" />  |
| <InlineMath math="90°" /> | <InlineMath math="1" />                  | <InlineMath math="1" />    | <InlineMath math="0" />                  | <InlineMath math="0" />    |

## Applications of Sine and Cosine in Real Life

Sine and cosine have many important applications in everyday life, especially in:

1. Measuring the height of buildings or objects
2. Navigation and direction finding
3. Architecture and construction
4. Physics and engineering
5. Design and calculation of structures

## Trigonometric Ratios in Pyramids

Let's look at an example of applying sine and cosine in the context of pyramids:

### Using Sine to Calculate Pyramid Height

Suppose an archaeologist wants to know the height of a pyramid. They know that the elevation angle from the base to the top of the pyramid is 41° and the slant height (edge) of the pyramid is 600 m.

<Triangle
  title="Calculating Pyramid Height with Sine"
  description="Triangle formed when calculating the height of a pyramid."
  angle={41}
  labels={{
    opposite: "Pyramid Height",
    adjacent: "Base Radius",
    hypotenuse: "Slant Height",
  }}
/>

To calculate the height of the pyramid, we use the sine ratio:

<div className="flex flex-col gap-4">
  <BlockMath math="\sin 41° = \frac{\text{pyramid height}}{\text{slant height}}" />
  <BlockMath math="\sin 41° = \frac{x \text{ m}}{600 \text{ m}}" />
  <BlockMath math="0.66 = \frac{x \text{ m}}{600 \text{ m}}" />
  <BlockMath math="x = 0.66 \times 600 \text{ m} = 396 \text{ m}" />
</div>

Therefore, the height of the pyramid is 396 meters.

### Using Cosine to Calculate Pyramid Base Radius

Now, if we want to know the base radius of the pyramid, we can use the cosine ratio:

<div className="flex flex-col gap-4">
  <BlockMath math="\cos 41° = \frac{\text{pyramid base radius}}{\text{slant height}}" />
  <BlockMath math="\cos 41° = \frac{x \text{ m}}{600 \text{ m}}" />
  <BlockMath math="0.75 = \frac{x \text{ m}}{600 \text{ m}}" />
  <BlockMath math="x = 0.75 \times 600 \text{ m} = 450 \text{ m}" />
</div>

Therefore, the base radius of the pyramid is 450 meters.

## Differences and Similarities Between Sin, Cos, and Tan

### Differences

1. **Sine (sin θ)** compares the opposite side with the hypotenuse.
2. **Cosine (cos θ)** compares the adjacent side with the hypotenuse.
3. **Tangent (tan θ)** compares the opposite side with the adjacent side.

### Similarities

1. All three are trigonometric ratios in right triangles.
2. All three change their values according to the angle.
3. These three ratios have a mathematical relationship:

   <BlockMath math="\tan \theta = \frac{\sin \theta}{\cos \theta}" />

## Practice Problem

A child is flying a kite and has managed to raise it to a height of 3.5 m. The child is holding the string at a height of 60 cm from the ground. If the kite string forms an angle of 25° with the ground, what is the length of the string being used?

<Triangle
  title="Kite Problem"
  description="Visualization of the kite string length problem."
  angle={25}
  labels={{
    opposite: "Kite Height",
    adjacent: "Horizontal Distance",
    hypotenuse: "String Length",
  }}
/>

To solve this problem, which trigonometric ratio should we use?

**Correct Solution:**

1. We need to calculate the string length (hypotenuse)
2. We know the effective height of the kite (3.5 m - 0.6 m = 2.9 m)
3. We know the elevation angle (25°)
4. Since we're looking for the hypotenuse and we know the opposite side (effective height), we use the sine ratio:

   <div className="flex flex-col gap-4">
     <BlockMath math="\sin 25° = \frac{\text{effective height}}{\text{string length}}" />
     <BlockMath math="\sin 25° = \frac{2.9 \text{ m}}{x \text{ m}}" />
     <BlockMath math="0.42 = \frac{2.9 \text{ m}}{x \text{ m}}" />
     <BlockMath math="x = \frac{2.9 \text{ m}}{0.42} = 6.9 \text{ m}" />
   </div>

Therefore, the length of the kite string being used is approximately 6.9 meters.
