Question Statement

(a) A conveyor belt in a factory is inclined at an angle of $18^{\circ }$. If the conveyor belt needs to move goods to a height of 6 meters, find the horizontal distance covered by the conveyor belt using the inverse tangent function.

(b) A lighthouse is 120 meters tall, and a ship is spotted at a distance of 250 meters from the base of the lighthouse. Find the angle of depression from the lighthouse to the ship using the inverse tangent function.

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Background and Explanation

Trigonometry relates the angles and sides of right-angled triangles. The tangent ratio ($\tan \theta =\frac{\text{Opposite}}{\text{Adjacent}}$) and its inverse ($\theta ={\tan }^{-1}(\frac{\text{Opposite}}{\text{Adjacent}})$) are used to find missing horizontal/vertical distances or angles of inclination and depression.

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Solution

Part (a)

In this problem, we represent the conveyor belt setup as a right-angled triangle. Let:

• $AC$ = The conveyor belt (hypotenuse).
• $AB$ = The horizontal length of the belt = $x$.
• $BC$ = The height to which the belt reaches = $6\text{ m}$.
• $\theta$ = The angle of inclination = $18^{\circ }$.

Using the relationship between the angle, the height (opposite), and the horizontal distance (adjacent):

$\begin{array}{rl}\theta & ={\tan }^{-1}\left(\frac{BC}{AB}\right)\\ 18^{\circ }& ={\tan }^{-1}\left(\frac{6}{x}\right)\\ \tan 18^{\circ }& =\frac{6}{x}\\ x\tan 18^{\circ }& =6\\ x& =\frac{6}{\tan 18^{\circ }}\\ x& =18.47\text{ m}\end{array}$

The horizontal distance covered by the conveyor belt is $18.47\text{ m}$.

Part (b)

We represent the lighthouse and the ship's position as a right-angled triangle.

Let:

• $AB$ = Height of the lighthouse = $120\text{ m}$.
• $AC$ = Distance of the ship from the base of the lighthouse = $250\text{ m}$.
• $\theta$ = The angle of depression (which is equal to the angle of elevation from the ship to the top of the lighthouse).

Using the inverse tangent function to find the angle:

$\begin{array}{rl}\theta & ={\tan }^{-1}\left(\frac{AB}{AC}\right)\\ & ={\tan }^{-1}\left(\frac{120}{250}\right)\\ \theta & =25^{\circ }38^{\prime }\end{array}$

The angle of depression from the lighthouse to the ship is $25^{\circ }38^{\prime }$.

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Key Formulas or Methods Used

• Tangent Ratio: $\tan \theta =\frac{\text{Opposite}}{\text{Adjacent}}$
• Inverse Tangent Function: $\theta ={\tan }^{-1}\left(\frac{\text{Opposite}}{\text{Adjacent}}\right)$
• Angle of Depression: The angle from the horizontal downward to an object, which is numerically equal to the angle of elevation from the object upward to the observer.

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Summary of Steps

1. Identify the knowns: Determine which sides (opposite, adjacent) or angles are provided in the problem.
2. Set up the equation: Use the tangent ratio if finding a side, or the inverse tangent function if finding an angle.
3. Substitute values: Plug the given measurements into the formula.
4. Solve for the unknown: Use algebraic manipulation to isolate the variable.
5. Calculate: Use a scientific calculator to find the final numerical value (ensuring the calculator is in Degree mode).