**Angles of elevation & depression are two types of angles that help us describe the relationship between an observer and an object.**

They are commonly used in trigonometry and geometry to measure the angle between a horizontal line and the line of sight from an observer to an object.

Angles of elevation & depression have real-world applications that can help us understand and navigate our surroundings.

We will explore what angles of elevation & depression are, how to calculate them, and provide you with some worked examples to solidify your understanding.

**An angle of elevation** is the angle formed when an observer looks upwards from a horizontal line to see an object that is above the observer’s eye level. Think of it as looking up at a tall building or a bird flying overhead.

On the other hand, **an angle of depression** is the angle formed when an observer looks downwards from a horizontal line to see an object that is below the observer’s eye level. This could be as simple as looking down at your feet or observing a boat on the water from a bridge.

#### Calculating Angles of Elevation & Depression

To calculate angles of elevation & depression, we need to have certain information at hand. Typically, we require the height or depth of the object being observed and the distance between the observer and the object.

Let’s take a look at a worked example to illustrate how to calculate these angles:

**Example 1:**

Ade is standing 50 meters away from a tall tree. If Ade’s eye level is 1.5 meters above the ground, and he looks up to see the top of the tree, what is the angle of elevation?

**Solution**

To solve this problem, we can use the tangent function, which relates the angle of elevation to the opposite and adjacent sides of a right triangle. – SOH CAH TOA.

First, we need to find the height of the tree above John’s eye level. This can be calculated by subtracting John’s eye level from the height of the tree. In this case, the height of the tree above John’s eye level is 50 meters – 1.5 meters = 48.5 meters.

Next, we can calculate the angle of elevation using the tangent function:

Angle of Elevation = arctan(opposite/adjacent) = arctan(48.5/50) ≈ 44.42°

Therefore, the angle of elevation is approximately 44.42°. This means that John needs to tilt his head upwards at an angle of 44.42° to see the top of the tree.

**Example 2:**

Mary is standing on a cliff that is 100 meters above sea level. She looks down and sees a boat on the water. If the boat is 150 meters away from the base of the cliff, what is the angle of depression?

**Solution**

Similar to the previous example, we can use the tangent function to calculate the angle of depression.

First, we need to find the depth of the boat below Sarah’s eye level. This can be calculated by subtracting the height of the cliff from the distance between the boat and the base of the cliff. In this case, the depth of the boat below Sarah’s eye level is 100 meters – 0 meters = 100 meters.

Next, we can calculate the angle of depression using the tangent function:

Angle of Depression = arctan(opposite/adjacent) = arctan(100/150) ≈ 33.69°

Therefore, the angle of depression is approximately 33.69°. This means that Sarah needs to tilt her head downwards at an angle of 33.69° to see the boat on the water from the cliff.

#### Conclusion

Angles of elevation & depression are important concepts that help us understand the relationship between an observer and an object. By knowing how to calculate these angles, we can gain a better understanding of our surroundings and make accurate observations.

Remember, the tangent function is your friend when it comes to calculating these angles. With a little practice and some basic trigonometry, you’ll be able to master angles of elevation and depression in no time.

So, the next time you find yourself looking up at a skyscraper or down at a beautiful landscape, take a moment to appreciate the angles involved and the mathematical beauty behind them.

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