Answered

In which triangle is the value of $x$ equal to $\tan^{-1}\left(\frac{3.1}{5.2}\right)$?

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Verified Answer :

  • The equation \(x = \tan^{-1}\left(\frac{3.1}{5.2}\right)\) identifies a right-angled triangle by the definition of the tangent function.
  • In such a triangle, the ratio of the side opposite angle \(x\) to the side adjacent to angle \(x\) must be exactly \(\frac{3.1}{5.2}\).
  • Consequently, the triangle is any right-angled triangle with leg lengths of \(3.1k\) and \(5.2k\) for any positive scalar \(k\).
  • The final answer is: \(\boxed{\text{A right-angled triangle with legs in the ratio } 3.1:5.2}\).

Explanation

  1. Understanding the definition of the inverse tangent function

The expression \(x = \tan^{-1}\left(\frac{3.1}{5.2}\right)\) involves the inverse tangent function, which is fundamental in trigonometry. In a right-angled triangle, 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. Given this, the equation \(x = \tan^{-1}\left(\frac{3.1}{5.2}\right)\) implies that there exists a right-angled triangle where one of the non-right angles, let's call it \(x\), satisfies \(\tan(x) = \frac{3.1}{5.2}\).

  1. Relating the equation to the properties of a right triangle

In a right-angled triangle with legs of lengths \(a\) and \(b\), the angle \(x\) opposite to leg \(a\) has a tangent defined as \(\tan(x) = \frac{a}{b}\). By comparing this with the given equation \(\tan(x) = \frac{3.1}{5.2}\), we can conclude that for any positive scaling factor \(k\), a right-angled triangle with leg lengths of \(3.1k\) and \(5.2k\) will have an angle \(x\) equal to \(\tan^{-1}\left(\frac{3.1}{5.2}\right)\). Specifically, the leg opposite the angle \(x\) has length \(3.1k\), and the leg adjacent to the angle \(x\) has length \(5.2k\).

  1. Conclusion on the characteristics of the triangle

Any right-angled triangle where the ratio of the two legs is \(3.1:5.2\) satisfies the given condition. The angle \(x\) will always be the angle opposite the leg of relative length \(3.1\). The other non-right angle in the triangle will be \(90^\circ - x = \tan^{-1}\left(\frac{5.2}{3.1}\right)\). Therefore, the triangle must be a right-angled triangle with specific side proportions.

Examples

Consider a landscape architect designing a ramp for a wheelchair-accessible entrance. If the building code requires the angle of inclination of the ramp to be \(\theta = \tan^{-1}\left(\frac{3.1}{5.2}\right)\), the architect can use this relationship to determine the required base length. By setting the rise (opposite side) to \(3.1\) meters and the run (adjacent side) to \(5.2\) meters, the architect creates a right-angled ramp structure that perfectly matches the required slope of \(\theta \approx 30.8^\circ\).

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