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URL: https://nakafa.com/en/subjects/mathematics/trigonometry/trigonometric-comparison-sin-cos
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Learn sine and cosine ratios with real-world pyramid examples. Compare trigonometric functions, solve practical problems, and understand their applications.
---
## What is the Sine Ratio?
Sine of an angle $$\theta$$ in a right triangle is the ratio between the length of the opposite side and the hypotenuse.
Visible text: Sine of an angle in a right triangle is the ratio between the length of the opposite side and the hypotenuse.
```math
\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}
```
Component: ContentBlock
Children:
Component: Triangle
Props:
- title: Visualization of Sine ($$\sin \theta$$)
Visible text: Visualization of Sine ()
- description: Move the slider to see how $$\sin$$ changes as the angle changes.
Visible text: Move the slider to see how changes as the angle changes.
- angle: 30
- labels: {
opposite: "Opposite Side",
adjacent: "Adjacent Side",
hypotenuse: "Hypotenuse",
}
### What is the Cosine Ratio?
Cosine of an angle $$\theta$$ in a right triangle is the ratio between the length of the adjacent side and the hypotenuse.
Visible text: Cosine of an angle in a right triangle is the ratio between the length of the adjacent side and the hypotenuse.
```math
\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}
```
Component: ContentBlock
Children:
Component: Triangle
Props:
- title: Visualization of Cosine ($$\cos \theta$$)
Visible text: Visualization of Cosine ()
- description: Move the slider to see how $$\cos$$ changes as the angle changes.
Visible text: Move the slider to see how changes as the angle changes.
- angle: 60
- labels: {
opposite: "Opposite Side",
adjacent: "Adjacent Side",
hypotenuse: "Hypotenuse",
}
## Sine and Cosine Values for Common Angles
Here are some sine and cosine values for commonly used angles:
| Angle | Sine Value $$\sin \theta$$ | Decimal Value | Cosine Value $$\cos \theta$$ | Decimal Value |
| ------------------------- | ---------------------------------------- | -------------------------- | ---------------------------------------- | -------------------------- |
| $$0^\circ$$ | $$0$$ | $$0$$ | $$1$$ | $$1$$ |
| $$30^\circ$$ | $$\frac{1}{2}$$ | $$0.5$$ | $$\frac{\sqrt{3}}{2}$$ | $$0.87$$ |
| $$45^\circ$$ | $$\frac{\sqrt{2}}{2}$$ | $$0.71$$ | $$\frac{\sqrt{2}}{2}$$ | $$0.71$$ |
| $$60^\circ$$ | $$\frac{\sqrt{3}}{2}$$ | $$0.87$$ | $$\frac{1}{2}$$ | $$0.5$$ |
| $$90^\circ$$ | $$1$$ | $$1$$ | $$0$$ | $$0$$ |
Visible text: | Angle | Sine Value | Decimal Value | Cosine Value | Decimal Value |
| ------------------------- | ---------------------------------------- | -------------------------- | ---------------------------------------- | -------------------------- |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
## 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 $$41^\circ$$ and the slant height (edge) of the pyramid is $$600 \text{ m}$$.
Visible text: 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 and the slant height (edge) of the pyramid is .
Component: Triangle
Props:
- 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:
Component: MathContainer
Children:
```math
\sin 41^\circ = \frac{\text{pyramid height}}{\text{slant height}}
```
```math
\sin 41^\circ = \frac{x \text{ m}}{600 \text{ m}}
```
```math
0.66 = \frac{x \text{ m}}{600 \text{ m}}
```
```math
x = 0.66 \times 600 \text{ m} = 396 \text{ m}
```
Therefore, the height of the pyramid is $$396 \text{ m}$$.
Visible text: Therefore, the height of the pyramid is .
### 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:
Component: MathContainer
Children:
```math
\cos 41^\circ = \frac{\text{pyramid base radius}}{\text{slant height}}
```
```math
\cos 41^\circ = \frac{x \text{ m}}{600 \text{ m}}
```
```math
0.75 = \frac{x \text{ m}}{600 \text{ m}}
```
```math
x = 0.75 \times 600 \text{ m} = 450 \text{ m}
```
Therefore, the base radius of the pyramid is $$450 \text{ m}$$.
Visible text: Therefore, the base radius of the pyramid is .
## Differences and Similarities Between Sin, Cos, and Tan
### Differences
1. **Sine** $$\sin \theta$$ compares the opposite side with the hypotenuse.
2. **Cosine** $$\cos \theta$$ compares the adjacent side with the hypotenuse.
3. **Tangent** $$\tan \theta$$ compares the opposite side with the adjacent side.
Visible text: 1. **Sine** compares the opposite side with the hypotenuse.
2. **Cosine** compares the adjacent side with the hypotenuse.
3. **Tangent** 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:
```math
\tan \theta = \frac{\sin \theta}{\cos \theta}
```
Visible text: 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:
## Practice Problem
A child is flying a kite and has managed to raise it to a height of $$3.5 \text{ m}$$. The child is holding the string at a height of $$60 \text{ cm}$$ from the ground. If the kite string forms an angle of $$25^\circ$$ with the ground, what is the length of the string being used?
Visible text: A child is flying a kite and has managed to raise it to a height of . The child is holding the string at a height of from the ground. If the kite string forms an angle of with the ground, what is the length of the string being used?
Component: Triangle
Props:
- 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, which is $$3.5 \text{ m} - 0.6 \text{ m} = 2.9 \text{ m}$$.
3. We know the elevation angle $$25^\circ$$.
4. Since we're looking for the hypotenuse and we know the opposite side (effective height), we use the sine ratio:
```math
\sin 25^\circ = \frac{\text{effective height}}{\text{string length}}
```
```math
\sin 25^\circ = \frac{2.9 \text{ m}}{x \text{ m}}
```
```math
0.42 = \frac{2.9 \text{ m}}{x \text{ m}}
```
```math
x = \frac{2.9 \text{ m}}{0.42} = 6.9 \text{ m}
```
Visible text: 1. We need to calculate the string length (hypotenuse)
2. We know the effective height of the kite, which is .
3. We know the elevation angle .
4. Since we're looking for the hypotenuse and we know the opposite side (effective height), we use the sine ratio:
Therefore, the length of the kite string being used is approximately $$6.9 \text{ m}$$.
Visible text: Therefore, the length of the kite string being used is approximately .