from the top of a vertical cliff. 65 metres high a lifeguard can see a boat out at sea the boat is 42 metres from the base of the cliff. What is the angle of depression of the boat from the top of the cliff?
draw a diagram. It should be clear that
tanθ = 65/42
To find the angle of depression, we can consider the right triangle formed by the vertical cliff, the line of sight from the lifeguard to the boat, and the horizontal distance from the base of the cliff to the boat.
Let's denote the angle of depression as θ.
From the given information, we have:
Opposite side = Height of the cliff = 65 meters
Adjacent side = Horizontal distance = 42 meters
Using the tangent function, we can find the angle of depression:
tan(θ) = Opposite side / Adjacent side
tan(θ) = 65 / 42
Now we can calculate the angle of depression:
θ = arctan(65/42)
θ ≈ 56.38 degrees
Therefore, the angle of depression of the boat from the top of the cliff is approximately 56.38 degrees.
To find the angle of depression, we need to visualize the situation and then use trigonometry. The angle of depression is the angle formed between the horizontal line (line of sight of the lifeguard looking straight ahead) and the line from the lifeguard's eye to the boat.
First, draw a diagram to help visualize the problem. Draw a vertical line to represent the cliff that is 65 meters high. From the base of the cliff, draw a horizontal line to represent the distance of 42 meters to the boat out at sea. Then, connect the top of the cliff to the boat, creating a right triangle.
Now, we can use the properties of right triangles and trigonometry. Remember that the tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
In this case, the opposite side is the height of the cliff (65 meters) and the adjacent side is the horizontal distance from the cliff to the boat (42 meters). We can use the formula for the tangent of an angle:
tan(angle) = opposite/adjacent
Plugging in the values, we get:
tan(angle) = 65/42
Now, to find the angle itself, we can use the inverse tangent function (also known as arctan or tan^(-1)):
angle = arctan(tan(angle))
Calculating this expression will give us the angle of depression. Using a calculator or a trigonometric table, you can find that the angle is approximately 56.29 degrees.
Therefore, the angle of depression of the boat from the top of the cliff is approximately 56.29 degrees.