TAN

Application:

One application of the tangent function is calculating the height of a tall object, such as a building or a tree, when you can't measure it directly. This is a common application in surveying and engineering.

Here's an example:

Imagine you are standing 100 feet away from the base of a flagpole. Using a transit or a clinometer, you measure the angle of elevation to the top of the flagpole to be 35∘. You want to find the height of the flagpole.

In this scenario, we can form a right-angled triangle where:

• The adjacent side (a) is the distance from you to the base of the flagpole (100 feet).
• The opposite side (o) is the height of the flagpole (h), which is what you want to find.
• The angle (θ) is the angle of elevation (35∘).

The tangent function is defined as:

tan(θ)=opposite/adjacent​

Plugging in the values from our example:

tan(35°)=100h​

To solve for h, we can rearrange the equation:

h=100×tan(35°)

Using a calculator, we find that tan(35°)≈0.7002.

h=100×0.7002

h≈70.02 feet

So, the height of the flagpole is approximately 70.02 feet.

This example illustrates how the tangent function allows us to relate the angle of a right-angled triangle to the ratio of its opposite and adjacent sides, making it possible to calculate unknown lengths.

Here is a table summarizing the relationship for this example: