Certainly! Let's solve the problem step-by-step.
### Given Data:
1. Height of the lighthouse (opposite side, in trigonometric terms) = 30 meters
2. Horizontal distance from the base of the lighthouse to the sea (adjacent side) = 30 meters
3. Angle of depression = 75 degrees
4. [tex]\(\tan(75^\circ) = 3.73\)[/tex]
### Desired:
We need to find the distance to the boat from the base of the lighthouse, using the given tangent value.
### Step-by-Step Solution:
1. Understanding the Problem:
- The height of the lighthouse forms the vertical leg of a right triangle.
- The horizontal distance forms the adjacent side of the right triangle.
- The angle of depression from the top of the lighthouse to the boat on the sea is given as 75 degrees.
- Using [tex]\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)[/tex], we need to find the distance between the lighthouse and the boat.
2. Trigonometric Identity:
We use the tangent of the angle of depression. By definition,
[tex]\[
\tan(75^\circ) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Given values, substitute them into the equation:
[tex]\[
3.73 = \frac{\text{Height of lighthouse}}{\text{Distance to boat}}
\][/tex]
3. Substitute Known Values:
- The height of the lighthouse is 30 meters.
- Let [tex]\(D\)[/tex] be the distance from the lighthouse to the boat.
[tex]\[
3.73 = \frac{30}{D}
\][/tex]
4. Solving for Distance [tex]\(D\)[/tex]:
Rearrange the equation to solve for [tex]\(D\)[/tex]:
[tex]\[
D = \frac{30}{3.73}
\][/tex]
5. Calculate [tex]\(D\)[/tex]:
Perform the division:
[tex]\[
D \approx 8.04289544235925 \text{ meters}
\][/tex]
### Conclusion:
Therefore, the distance between the boat and the house (lighthouse) is approximately 8.04 meters.
