Question

find the missing side.
13
33°
y
y = ?

Explanation:

Step1: Identify trigonometric ratio

We have a right triangle with angle \(33^\circ\), opposite side \(13\), and adjacent side \(y\). So we use \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), but wait, no—wait, \(\tan(33^\circ)=\frac{13}{y}\)? Wait, no, wait: the angle is \(33^\circ\), the side opposite is \(13\), and the adjacent is \(y\)? Wait, no, wait, in the right triangle, the right angle is at the bottom right, so the angle at the bottom left is \(33^\circ\), the vertical side is \(13\) (opposite to \(33^\circ\)? No, wait, no: the horizontal side is \(y\) (adjacent), vertical side is \(13\) (opposite to the \(33^\circ\) angle? Wait, no, \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). So \(\tan(33^\circ)=\frac{13}{y}\)? Wait, no, wait, maybe I got opposite and adjacent wrong. Wait, the angle is \(33^\circ\), the side opposite to \(33^\circ\) is the vertical side (length 13), and the adjacent side is the horizontal side \(y\). So \(\tan(33^\circ)=\frac{\text{opposite}}{\text{adjacent}}=\frac{13}{y}\)? Wait, no, that would mean \(y = \frac{13}{\tan(33^\circ)}\). Wait, or is it \(\cot(33^\circ)=\frac{y}{13}\), since \(\cot\theta=\frac{\text{adjacent}}{\text{opposite}}\). So \(\cot(33^\circ)=\frac{y}{13}\), so \(y = 13 \times \cot(33^\circ)\), or \(y = \frac{13}{\tan(33^\circ)}\).

Step2: Calculate the value

First, find \(\tan(33^\circ)\). Using a calculator, \(\tan(33^\circ) \approx 0.6494\). Then \(y = \frac{13}{0.6494} \approx 19.9\) (or more accurately, let's compute it properly). Wait, let's check: \(\tan(33^\circ) \approx 0.6494075931\). So \(y = \frac{13}{0.6494075931} \approx 19.9\) (or 20 when rounded, but let's do it more accurately). Alternatively, using \(\cot(33^\circ) = \frac{1}{\tan(33^\circ)} \approx 1.5399\), so \(y = 13 \times 1.5399 \approx 20.0187\), so approximately 20. Wait, maybe I made a mistake in opposite and adjacent. Wait, no: the angle is at the bottom left, so the sides: the side opposite to \(33^\circ\) is the vertical side (13), adjacent is the horizontal side (y). So \(\tan(33^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{13}{y}\), so solving for \(y\): \(y = \frac{13}{\tan(33^\circ)}\). Let's compute that. \(\tan(33^\circ) \approx 0.6494\), so \(13 / 0.6494 \approx 19.9\), which is approximately 20. Wait, but maybe it's \(\cot\), but \(\cot\) is adjacent over opposite, so \(\cot(33^\circ) = y / 13\), so \(y = 13 \times \cot(33^\circ)\), and \(\cot(33^\circ) = 1 / \tan(33^\circ) \approx 1.5399\), so \(13 \times 1.5399 \approx 20.0187\), so \(y \approx 20\) (or more precise, let's use calculator values). Let's use a calculator for \(\tan(33^\circ)\):

\(\tan(33^\circ) \approx 0.6494075931\)

So \(y = 13 / 0.6494075931 \approx 19.9\) (or 20 when rounded to a reasonable decimal place). Wait, maybe the problem expects using tangent or cotangent. Let's confirm: in a right triangle, for angle \(\theta\), \(\tan\theta = \text{opposite}/\text{adjacent}\), \(\cot\theta = \text{adjacent}/\text{opposite}\), \(\sin\theta = \text{opposite}/\text{hypotenuse}\), \(\cos\theta = \text{adjacent}/\text{hypotenuse}\). So here, angle is \(33^\circ\), opposite side is 13, adjacent is \(y\), so \(\tan(33^\circ) = 13 / y\) => \(y = 13 / \tan(33^\circ)\). Let's compute that:

\(13 \div \tan(33^\circ) \approx 13 \div 0.6494 \approx 19.9\) (or 20.0 when rounded to one decimal place, or 20 as a whole number).

Answer:

\(y \approx 20\) (or more precisely, approximately 19.9 or 20.0, depending on rounding)