23. A man is in a boat that is floating 175 feet from the base of a 200 foot cliff. What is the angle of depression between the cliff and the boat?

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To find the angle of depression between the cliff and the boat, we can use the concept of trigonometry.

Given:
- The height of the cliff is 200 feet.
- The distance from the base of the cliff to the boat is 175 feet.

We can use the tangent function to find the angle of depression. The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Let's denote the angle of depression as θ (theta).

Using the tangent function:
$$ \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} $$

In this case, the opposite side is the height of the cliff (200 feet), and the adjacent side is the distance from the base of the cliff to the boat (175 feet).

$$ \tan(\theta) = \frac{200}{175} $$

Now, we can calculate the angle of depression by taking the inverse tangent (arctangent) of the ratio:

$$ \theta = \arctan\left(\frac{200}{175}\right) $$

Let's calculate the angle of depression.
Calculate the value by following steps:
- step0: Calculate:
  $$\arctan\left(\frac{200}{175}\right)$$
- step1: Reduce the fraction:
  $$\arctan\left(\frac{8}{7}\right)$$
The angle of depression between the cliff and the boat is approximately 0.851966 radians or 48.7 degrees.
The angle of depression between the cliff and the boat is approximately 48.7 degrees.

23. A man is in a boat that is floating 175 feet from the base of a 200 foot cliff. What is the angle of depression between the cliff and the boat?

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