Question Statement

A lighthouse is 90 meters tall, and a boat is observed at a distance of 180 meters from the base of the lighthouse. Determine the angle of elevation from the boat to the top of the lighthouse using the inverse tangent function.

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Background and Explanation

This problem uses right-angled trigonometry to find an unknown angle. The angle of elevation is the angle between the horizontal ground and the line of sight to the top of the lighthouse, which can be found using the ratio of the opposite side (height) to the adjacent side (distance).

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Solution

To solve this problem, we represent the scenario as a right-angled triangle where:

• $AB$ is the height of the lighthouse (the side opposite to the angle $\theta$).
• $AC$ is the distance of the boat from the base of the lighthouse (the side adjacent to the angle $\theta$).

Given values:

• $AB=\text{Height of lighthouse}=90\text{ m}$
• $AC=\text{Distance of boat from point }A=180\text{ m}$

We use the tangent ratio, which is defined as $\tan (\theta )=\frac{\text{Opposite}}{\text{Adjacent}}$. To find the angle $\theta$ itself, we use the inverse tangent function (${\tan }^{-1}$):

$\theta ={\tan }^{-1}\left(\frac{AB}{AC}\right)$

Substitute the known values into the equation:

$\theta ={\tan }^{-1}\left(\frac{90}{180}\right)$

Simplify the fraction inside the parentheses:

$\theta ={\tan }^{-1}(0.5)$

Using a calculator to find the inverse tangent of $0.5$:

$\theta =26^{\circ }34^{\prime }$

The angle of elevation from the boat to the top of the lighthouse is $26^{\circ }34^{\prime }$.

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Key Formulas or Methods Used

• Tangent Ratio: $\tan (\theta )=\frac{\text{Opposite}}{\text{Adjacent}}$
• Inverse Tangent Function: $\theta ={\tan }^{-1}\left(\frac{\text{Opposite}}{\text{Adjacent}}\right)$

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Summary of Steps

1. Identify the height of the lighthouse as the opposite side ($90\text{ m}$).
2. Identify the distance from the base as the adjacent side ($180\text{ m}$).
3. Apply the inverse tangent formula: $\theta ={\tan }^{-1}\left(\frac{\text{Opposite}}{\text{Adjacent}}\right)$.
4. Simplify the ratio and calculate the final angle in degrees and minutes.