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$X Y Z$ is a right triangle with right angle $Y$. If the measure of angle $X$ is $35^{\circ}$, write two trigonometric equations that express side length $x$ in terms of $y, z$ and the acute angle $X$.


A$x=z \cdot \cot \left(35^{\circ}\right) ; x=y \cdot \cos \left(35^{\circ}\right)$
B$x=z \cdot \tan \left(35^{\circ}\right) ; x=y \cdot \sec \left(35^{\circ}\right)$
C$x=z \cdot \cot \left(35^{\circ}\right) ; x=y \cdot \sin \left(35^{\circ}\right)$
D$x=z \cdot \tan \left(35^{\circ}\right) ; x=y \cdot \cos \left(35^{\circ}\right)$
E$x=z \cdot \tan \left(35^{\circ}\right) ; x=y \cdot \sin \left(35^{\circ}\right)$

F. None of the above

Verified Answer :

  • Define sides of right triangle \(XYZ\) based on angle \(X\): \(x\) is opposite, \(z\) is adjacent, \(y\) is hypotenuse.
  • Use \(\tan(X) = \frac{x}{z}\) to get \(x = z \cdot \tan(35^{\circ})\).
  • Use \(\sin(X) = \frac{x}{y}\) to get \(x = y \cdot \sin(35^{\circ})\).
  • Select the option: \(\boxed{x=z \cdot \tan(35^{\circ}) ; x=y \cdot \sin(35^{\circ})}\).

Explanation

  1. Analyze the triangle and identify sides.

In a right triangle \(XYZ\) with the right angle at \(Y\), the sides are denoted as \(x\) (the leg opposite vertex \(X\)), \(z\) (the leg adjacent to vertex \(X\)), and \(y\) (the hypotenuse, opposite the right angle \(Y\)). We are given that \(\angle X = 35^{\circ}\). We need to express the side length \(x\) using the trigonometric ratios defined by this angle.

  1. Derive trigonometric equations.

Using the definitions of trigonometric ratios in a right triangle:

  1. The tangent of an angle is the ratio of the opposite leg to the adjacent leg:

\(\tan(X) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{z}\)
Rearranging this to solve for \(x\), we get:
\(x = z \cdot \tan(X) = z \cdot \tan(35^{\circ})\)

  1. The sine of an angle is the ratio of the opposite leg to the hypotenuse:

\(\sin(X) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{y}\)
Rearranging this to solve for \(x\), we get:
\(x = y \cdot \sin(X) = y \cdot \sin(35^{\circ})\)

  1. Select the correct answer.

Comparing our derived equations \(x = z \cdot \tan(35^{\circ})\) and \(x = y \cdot \sin(35^{\circ})\) with the provided options:

  • \(x=z \cdot \cot \left(35^{\circ}\right) ; x=y \cdot \cos \left(35^{\circ}\right)\)
  • \(x=z \cdot \tan \left(35^{\circ}\right) ; x=y \cdot \sec \left(35^{\circ}\right)\)
  • \(x=z \cdot \cot \left(35^{\circ}\right) ; x=y \cdot \sin \left(35^{\circ}\right)\)
  • \(x=z \cdot \tan \left(35^{\circ}\right) ; x=y \cdot \cos \left(35^{\circ}\right)\)
  • \(x=z \cdot \tan \left(35^{\circ}\right) ; x=y \cdot \sin \left(35^{\circ}\right)\)

Our equations match the fifth option. Thus, the correct pair of equations is \(x=z \cdot \tan(35^{\circ})\) and \(x=y \cdot \sin(35^{\circ})\).

Examples

In architecture, if you need to calculate the height of a wall (the side \(x\)) relative to the ground distance (the adjacent side \(z\)) or the length of a support beam (the hypotenuse \(y\)) when the angle of elevation is \(35^{\circ}\), these trigonometric relationships are essential. For example, if you know the ground distance \(z = 10\) meters, the wall height is \(x = 10 \cdot \tan(35^{\circ}) \approx 7.00\) meters. If you know the beam length \(y = 12\) meters, the height is \(x = 12 \cdot \sin(35^{\circ}) \approx 6.88\) meters.

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