Certainly! Let's analyze the given table and observe the three calculated ratios: [tex]\(\frac{\text{opp}}{\text{hyp}}\)[/tex], [tex]\(\frac{\text{adj}}{\text{hyp}}\)[/tex], and [tex]\(\frac{\text{opp}}{\text{adj}}\)[/tex].
Here is the table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Triangle} & \text{Hypotenuse} & \text{Opposite} & \text{Adjacent} & \frac{\text{opp}}{\text{hyp}} & \frac{\text{adj}}{\text{hyp}} & \frac{\text{opp}}{\text{adj}} \\
\hline
1 & 2.742 & 0.4761433 & 2.7003429 & 0.1736481765134938 & 0.9848077680525165 & 0.17632697684431115 \\
\hline
2 & 2.742 & 0.70968182 & 2.6485686 & 0.25881904449307075 & 0.9659258205689278 & 0.2679491933869487 \\
\hline
3 & 2.742 & 0.93781923 & 2.5766372 & 0.3420201422319475 & 0.939692633114515 & 0.36397022832706133 \\
\hline
4 & 2.742 & 1.1588193 & 2.485096 & 0.42261827133479213 & 0.9063078045222466 & 0.4663076597443318 \\
\hline
5 & 2.742 & 1.5727466 & 2.2461149 & 0.5735764405543399 & 0.8191520423048869 & 0.7002075450369881 \\
\hline
\end{array}
\][/tex]
### Observations
1. [tex]\(\frac{\text{opp}}{\text{hyp}}\)[/tex]:
- For Triangle 1: [tex]\(\frac{0.4761433}{2.742} = 0.1736481765134938\)[/tex]
- For Triangle 2: [tex]\(\frac{0.70968182}{2.742} = 0.25881904449307075\)[/tex]
- For Triangle 3: [tex]\(\frac{0.93781923}{2.742} = 0.3420201422319475\)[/tex]
- For Triangle 4: [tex]\(\frac{1.1588193}{2.742} = 0.42261827133479213\)[/tex]
- For Triangle 5: [tex]\(\frac{1.5727466}{2.742} = 0.5735764405543399\)[/tex]
2. [tex]\(\frac{\text{adj}}{\text{hyp}}\)[/tex]:
- For Triangle 1: [tex]\(\frac{2.7003429}{2.742} = 0.9848077680525165\)[/tex]
- For Triangle 2: [tex]\(\frac{2.6485686}{2.742} = 0.9659258205689278\)[/tex]
- For Triangle 3: [tex]\(\frac{2.5766372}{2.742} = 0.939692633114515\)[/tex]
- For Triangle 4: [tex]\(\frac{2.485096}{2.742} = 0.9063078045222466\)[/tex]
- For Triangle 5: [tex]\(\frac{2.2461149}{2.742} = 0.8191520423048869\)[/tex]
3. [tex]\(\frac{\text{opp}}{\text{adj}}\)[/tex]:
- For Triangle 1: [tex]\(\frac{0.4761433}{2.7003429} = 0.17632697684431115\)[/tex]
- For Triangle 2: [tex]\(\frac{0.70968182}{2.6485686} = 0.2679491933869487\)[/tex]
- For Triangle 3: [tex]\(\frac{0.93781923}{2.5766372} = 0.36397022832706133\)[/tex]
- For Triangle 4: [tex]\(\frac{1.1588193}{2.485096} = 0.4663076597443318\)[/tex]
- For Triangle 5: [tex]\(\frac{1.5727466}{2.2461149} = 0.7002075450369881\)[/tex]
### Overall Trends
- The [tex]\(\frac{\text{opp}}{\text{hyp}}\)[/tex] ratio shows a gradual increase as we move from Triangle 1 to Triangle 5.
- The [tex]\(\frac{\text{adj}}{\text{hyp}}\)[/tex] ratio decreases as we move from Triangle 1 to Triangle 5.
- The [tex]\(\frac{\text{opp}}{\text{adj}}\)[/tex] ratio increases considerably from Triangle 1 to Triangle 5.
These observations suggest that:
- As the angle opposite to the "Opposite" side increases, the [tex]\(\frac{\text{opp}}{\text{hyp}}\)[/tex] ratio (which is essentially [tex]\(\sin(\theta)\)[/tex]) increases.
- Similarly, the [tex]\(\frac{\text{adj}}{\text{hyp}}\)[/tex] ratio, analogous to [tex]\(\cos(\theta)\)[/tex], decreases as the angle increases.
- The [tex]\(\frac{\text{opp}}{\text{adj}}\)[/tex] ratio, which corresponds to [tex]\(\tan(\theta)\)[/tex], increases significantly, showing a rapid rate of change as the angle increases.
