From the top of a cliff 120 m above the water, the angle of depression of a boat on the water is 18 degrees. How far is the boat from the cliff?

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To find the distance between the boat and the cliff, we can use trigonometric ratios and the angle of depression.

1. Draw a diagram: Start by drawing a line to represent the cliff, and label the top of the cliff as point A. Draw a horizontal line to represent the water, and label the point directly below point A as point B. Label the boat as point C.

2. Identify the angle of depression: The angle of depression is the angle between the horizontal line (AB) and the line of sight from the top of the cliff to the boat (AC). In this case, the angle of depression is given as 18 degrees.

3. Identify the right triangle: Drawing a line segment from point A to point C, we can create a right triangle with AC as the hypotenuse, AB as the horizontal side, and BC as the vertical side.

4. Identify the trigonometric ratio: In this case, we are given the angle of depression, which is the angle opposite the vertical side (BC) in the right triangle. Since we are trying to find the distance AC (hypotenuse), we will use the tangent ratio, which is defined as the opposite side divided by the adjacent side.

5. Apply the tangent ratio: The tangent of an angle theta is defined as the opposite side (BC) divided by the adjacent side (AB). Therefore, we can write the equation: tangent(18 degrees) = BC / AB.

6. Solve for BC: Rearranging the equation, BC = AB * tangent(18 degrees).

7. Calculate the distance: We are given the height of the cliff as 120 m, so the vertical side BC is equal to 120 m. Plugging in the value for BC and solving for AB, we have: 120 m = AB * tangent(18 degrees).

8. Calculate AB: Divide both sides of the equation by the tangent(18 degrees) to solve for AB. AB = 120 m / tangent(18 degrees).

9. Use a calculator: Use a calculator to find the value of tangent(18 degrees) and perform the calculation to find the value of AB.

10. Final answer: The distance between the boat and the cliff, AB, is equal to the value of AB calculated in step 9.

To find the distance between the boat and the cliff, we can make use of trigonometric functions, specifically the tangent function.

In this case, the angle of depression, which is the angle between the horizontal line and the line of sight from the top of the cliff to the boat, is given as 18 degrees.

The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Let's define the length of the side adjacent to the angle as the horizontal distance from the cliff to the boat, which is what we want to find. We can call this distance "x".

The length of the side opposite the angle is the vertical distance from the top of the cliff to the water, which is given as 120 m.

Using the tangent function, we have the following equation:

tan(18) = opposite/adjacent

Substituting the given values:

tan(18) = 120 / x

To solve for x, we can rearrange the equation:

x = 120 / tan(18)

Now we can calculate the value of x by plugging in the values into a calculator or a math software. The result will give us the distance between the boat and the cliff.

tan 18 = 120/x