Let's break down the problem and verify the given statements step-by-step:
### Given:
- Distance from the boat to the lighthouse: [tex]\( d = 110 \)[/tex] meters
- Angle of elevation: [tex]\( \theta = 44^\circ \)[/tex]
### Key mathematical relationships involving angles and sides:
- [tex]\(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)[/tex]
- [tex]\(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\)[/tex]
### Define:
- [tex]\( y \)[/tex]: The length of the adjacent side for the cosine function.
### Cosine of the angle:
According to the cosine function, we have:
[tex]\[ \cos 44^\circ = \frac{y}{110} \][/tex]
So:
[tex]\[ y = 110 \cdot \cos 44^\circ \][/tex]
Let’s examine each of the given statements one by one:
1. [tex]\(\cos 44^\circ = \frac{y}{110}\)[/tex]
By the definition of cosine:
[tex]\[ \cos 44^\circ = \frac{y}{110} \][/tex]
This is [tex]\(True\)[/tex] because it directly follows from the definition.
2. [tex]\(\cos 44^\circ = \frac{110}{y}\)[/tex]
This implies:
[tex]\[ y = 110 \cdot \cos 44^\circ \][/tex]
Rewriting [tex]\(\cos 44^\circ = \frac{110}{y}\)[/tex] would mean:
[tex]\[ y = \frac{110}{\cos 44^\circ} \][/tex]
This doesn't match our previous relation [tex]\(y = 110 \cdot \cos 44^\circ\)[/tex]. So, this statement is [tex]\(False\)[/tex].
3. [tex]\(\tan 44^\circ = \frac{y}{110}\)[/tex]
By definition:
[tex]\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
This would imply:
[tex]\[ \tan 44^\circ = \frac{y}{110} \][/tex]
But we know from trigonometric identities that:
[tex]\[ \tan 44^\circ \neq \frac{y}{110} \][/tex]
Hence, this statement is [tex]\(False\)[/tex].
4. [tex]\(\tan 44^\circ = \frac{110}{y}\)[/tex]
Similarly,
[tex]\[ \tan 44^\circ = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
So:
[tex]\[ \tan 44^\circ = \frac{110}{y} \][/tex]
Again, referring to the trigonometric functions, this doesn't align with our calculations:
[tex]\[112.178\neq \][/tex]
Hence, this statement is also [tex]\(False\)[/tex].
### Conclusion:
- [tex]\(\cos 44^\circ = \frac{y}{110}\)[/tex] is [tex]\(True\)[/tex].
- [tex]\(\cos 44^\circ = \frac{110}{y}\)[/tex] is [tex]\(False\)[/tex].
- [tex]\(\tan 44^\circ = \frac{y}{110}\)[/tex] is [tex]\(False\)[/tex].
- [tex]\(\tan 44^\circ = \frac{110}{y}\)[/tex] is [tex]\(False\)[/tex].
So, the overall results are:
[tex]\[(True, False, False, False)\][/tex]
