In right triangle XYZ, angle X is 22 degrees and XZ is 8. Find the length of YZ, rounded to the nearest tenth.

Question image

Understand the Problem

The question requires finding the length of side YZ in a right-angled triangle XYZ, given the length of side XZ and the angle at vertex X. We can use trigonometric ratios (specifically the tangent function) to solve for YZ.

Answer

$YZ = 3.2$
Answer for screen readers

$YZ = 3.2$

Steps to Solve

  1. Identify the relevant trigonometric ratio

Since we have the adjacent side (XZ) and want to find the opposite side (YZ) with respect to the given angle at X, we use the tangent function:

$tan(\theta) = \frac{opposite}{adjacent}$

  1. Plug in the given values

We are given that angle X is 22 degrees and the length of XZ is 8. So we have:

$tan(22^\circ) = \frac{YZ}{8}$

  1. Solve for YZ

Multiply both sides of the equation by 8:

$YZ = 8 \cdot tan(22^\circ)$

  1. Calculate the value

Using a calculator, we find that $tan(22^\circ) \approx 0.4040$. Thus,

$YZ \approx 8 \cdot 0.4040 = 3.232$

  1. Round to the nearest tenth

Rounding 3.232 to the nearest tenth gives us 3.2.

$YZ = 3.2$

More Information

The tangent function is a fundamental concept in trigonometry and is used to relate the angles of a right triangle to the ratios of its sides.

Tips

A common mistake is using the wrong trigonometric function (e.g., sine or cosine instead of tangent). Another mistake is not setting up the equation correctly, for instance, inverting the ratio of opposite to adjacent. Also, forgetting to round to the specified decimal place.

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