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Calculate tangent ratios using opposite and adjacent sides. Solve height problems, slopes, and angles with worked examples and practice exercises.

---

## 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 written as $$\tan$$.

Visible text: When studying trigonometry, we encounter several types of ratios. One of the most fundamental is the tangent ratio, often written as .

### Understanding Tangent

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.

```math
\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}
```

Component: ContentBlock
Children:
Component: Triangle
Props:
- title: Visualization of Tangent in a Triangle
- description: The $$\tan$$ ratio compares the opposite side to the adjacent side.
  Visible text: The ratio compares the opposite side to the adjacent side.
- angle: 30
- labels: {
opposite: "opposite side",
adjacent: "adjacent side",
hypotenuse: "hypotenuse",
}

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

- The opposite side is the side facing the angle $$\theta$$.
- The adjacent side is the side next to the angle $$\theta$$ other than the hypotenuse.
- Tangent is calculated by dividing the length of the opposite side by the adjacent side

Visible text: - The opposite side is the side facing the angle .
- The adjacent side is the side next to the angle 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:

- $$\tan 30^\circ = 0.57 \text{ or} \frac{1}{\sqrt{3}}$$
- $$\tan 45^\circ = 1$$
- $$\tan 60^\circ = 1.73 \text{ or} \sqrt{3}$$

Visible text: - 
- 
-

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 $$30^\circ$$, an opposite side of $$15 \text{ cm}$$, and an adjacent side of $$26 \text{ cm}$$.

Visible text: For instance, if we have a right-angled triangle with an angle of , an opposite side of , and an adjacent side of .

Component: Triangle
Props:
- title: Example Triangle with $$30^\circ$$ Angle
  Visible text: Example Triangle with Angle
- description: Ratio of opposite side ($$15 \text{ cm}$$) to adjacent
side ($$26 \text{ cm}$$).
  Visible text: Ratio of opposite side () to adjacent
side ().
- angle: 30
- labels: {
opposite: "15 cm",
adjacent: "26 cm",
hypotenuse: "hypotenuse",
}

The tangent value of this angle is:

```math
\tan 30^\circ = \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 $$70^\circ$$, an opposite side of $$15 \text{ cm}$$, and an adjacent side of $$7 \text{ cm}$$.

Visible text: Consider the following right-angled triangle with an angle of , an opposite side of , and an adjacent side of .

Component: Triangle
Props:
- title: Triangle with $$70^\circ$$ Angle
  Visible text: Triangle with Angle
- description: Triangle with opposite side $$15 \text{ cm}$$ and
adjacent side $$7 \text{ cm}$$.
  Visible text: Triangle with opposite side and
adjacent side .
- angle: 70
- labels: {
opposite: "15 cm",
adjacent: "7 cm",
hypotenuse: "hypotenuse",
}

Can you find the tangent ratio value of $$\tan 70^\circ$$? Explain why!

Visible text: Can you find the tangent ratio value of ? Explain why!

### Answer Key

Yes, we can find the value of $$\tan 70^\circ$$ by dividing the opposite side by the adjacent side:

Visible text: Yes, we can find the value of by dividing the opposite side by the adjacent side:

```math
\tan 70^\circ = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{15 \text{ cm}}{7 \text{ cm}} = 2.14
```

Therefore, the value of $$\tan 70^\circ = 2.14$$.

Visible text: Therefore, the value of .