Question Statement

A robotic arm on an assembly line needs to reach a point 4 meters away (horizontally) and 3 meters above its base (vertically). Find the angle $\theta$ at which the arm should be positioned relative to the horizontal using the inverse tangent function.

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Background and Explanation

This problem involves right-angled trigonometry. When we know the vertical height (opposite side) and the horizontal distance (adjacent side) of a point relative to an origin, we can use the tangent ratio to find the angle of elevation.

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Solution

To find the angle of the robotic arm, we model the situation as a right-angled triangle where the arm forms the hypotenuse.

1. Identify the given dimensions: Based on the problem description:

• Height of the robotic arm ($AB$) = $3\text{ m}$ (This is the perpendicular side)
• Width/Horizontal distance ($AC$) = $4\text{ m}$ (This is the base side)
• The angle to find is $\angle ACB=\theta$

2. Apply the tangent ratio: The tangent of an angle in a right triangle is the ratio of the perpendicular side to the base side: $\tan \theta =\frac{\text{Perpendicular}}{\text{Base}}$

Substituting the known values: $\tan \theta =\frac{3}{4}$

3. Solve for $\theta$ using the inverse tangent function: To isolate $\theta$, we take the inverse tangent (${\tan }^{-1}$) of both sides: $\theta ={\tan }^{-1}\left(\frac{3}{4}\right)$

4. Calculate the final value: Using a calculator to find the inverse tangent of $0.75$: $\theta \approx 36.87^{\circ }$

The robotic arm should be positioned at an angle of approximately $36.87^{\circ }$.

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Key Formulas or Methods Used

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

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Summary of Steps

1. Identify the vertical height ($3\text{ m}$) and horizontal distance ($4\text{ m}$).
2. Set up the trigonometric ratio $\tan \theta =\frac{\text{height}}{\text{distance}}$.
3. Apply the inverse tangent function to solve for the angle: $\theta ={\tan }^{-1}(3/4)$.
4. Calculate the numerical value to determine the positioning angle.