Trigonometric Comparison: Tan θ

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.

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:

• $\tan 30° = 0.57 \text{ or } \frac{1}{\sqrt{3}}$
• $\tan 45° = 1$
• $\tan 60° = 1.73 \text{ or } \sqrt{3}$

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 $30°$, an opposite side of 15 cm, and an adjacent side of 26 cm.

The tangent value of this angle is:

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 $70°$, an opposite side of 15 cm, and an adjacent side of 7 cm.

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

Answer Key

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

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