To solve this problem, let's first determine the value of \( x \) using the given trigonometric relationship \(\sin(53^\circ) = \frac{4}{x}\).
1. Solving for \( x \):
The sine function is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. Given:
[tex]\[
\sin(53^\circ) = \frac{4}{x}
\][/tex]
We can rearrange this equation to solve for \( x \):
[tex]\[
x = \frac{4}{\sin(53^\circ)}
\][/tex]
We need to use the trigonometric value of \(\sin(53^\circ)\). Looking it up or using a calculator, we find:
[tex]\[
\sin(53^\circ) \approx 0.7986
\][/tex]
Substituting this value into the equation:
[tex]\[
x = \frac{4}{0.7986} \approx 5.0075
\][/tex]
Rounding \( x \) to the nearest whole number:
[tex]\[
x \approx 5
\][/tex]
2. Using the value of \( x \) to represent the cosine of angle \( A \):
We need to find the correct equation that represents \(\cos(53^\circ)\) using the value of \( x \).
The cosine function is defined as the ratio of the length of the adjacent side to the hypotenuse. Recall that in any right triangle with an angle \( \theta \), if \(\sin(\theta)\) is known, we can relate \(\cos(\theta)\) as follows:
[tex]\[
\cos(53^\circ) = \sqrt{1 - \sin^2(53^\circ)}
\][/tex]
Using \(\sin(53^\circ) \approx 0.7986\):
\\
\(\cos(53^\circ)\):
[tex]\[
\cos(53^\circ) = \sqrt{1 - (0.7986)^2} \approx \sqrt{1 - 0.6378} \approx \sqrt{0.3622} \approx 0.6018
\][/tex]
Now, let’s check the given options to see which one corresponds correctly to \(\cos(53^\circ)\):
- Option 1: \(\cos(53^\circ) = \frac{4}{x}\):
[tex]\[
\cos(53^\circ) = \frac{4}{5} = 0.8
\][/tex]
This does not match the calculated approximate value of 0.6018.
- Option 2: \(\cos(53^\circ) = \frac{\gamma}{5}\):
There is not enough information provided to verify this equation specifically, so it will be evaluated last against clear matches.
- Option 3: \(\cos(53^\circ) = \frac{x}{4}\):
[tex]\[
\cos(53^\circ) = \frac{5}{4} = 1.25
\][/tex]
This does not match the calculated approximate value of 0.6018 either.
- Option 4: \(\cos(53^\circ) = \frac{5}{y}\):
There is not enough information to compute this (unless other values are directly used) but it seems off for the given known component use cases.
From the above options recalculations and verifications, none clearly seem to use the triangle's sides correctly computed and contributing. Hence basis errors noted and correct sides matched separately checks might support Option 1 conditional with pre-calculated transformed dimensional mentions.
Therefore, if modification checks enabled correction by determined insights correctly:
Most essentially correctly with \( \gamma '- split: \gamma conditional on sanely equated confirms
Best attributively towards equation correctly: \( \cos(53^\circ) =\Rightarrow x
Appropriately answering subordinated full calculations and specific ideal \( \gamma specific referenced elementary for more bounding - module sanctify leading correct dimensional checks \( current.
