A rectangular park had a dirt path across its diagonal that was yards long. The diagonal and the long side of the park formed an angle that measured . A person walked along the sidewalks outside the park, from the start to the end of the path, as shown by the arrows. The diagonal dirt path that cuts across the rectangular park and the sidewalks outside the park form a right triangle. The dirt path is the hypotenuse and it is labeled 100 yards. The start of the sidewalk is adjacent to the 30-degree angle, and it is labeled X. The end of the path is opposite the 30-degree angle, and it is labeled Y. Which expression shows the distance that he walked?

Trigonometry functions are functions that relate two sides of a given triangle with one of its included angles. The required function may be a sine function, a cosine function, or a tangent function.

Such that;

Sin θ = [tex]\frac{Opposite}{Hypotenuse}[/tex]

Cos θ = [tex]\frac{Adjacent}{Hypotenuse}[/tex]

Tan θ = [tex]\frac{Opposite}{Adjacent}[/tex]

The given question can be solved by applying the appropriate trigonometric function.

So that to determine the value of x, we have:

Cos θ = [tex]\frac{Adjacent}{Hypotenuse}[/tex]

Cos 30 = [tex]\frac{x}{100}[/tex]

⇒ x = 100 cos 30

= 100 x 0.866

x = 86.60

Thus, the start of the sidewalk is approximately 87 yards.

To determine the value of sidewalk y, we have:

Sin θ = [tex]\frac{Opposite}{Hypotenuse}[/tex]

Sin 30 = [tex]\frac{y}{100}[/tex]

⇒ y = 100 x Sin 30

= 100 x 0.5

y = 50

Thus the sidewalk which represents the end of the path is 50 yards.

Therefore, the total distance that he walked = 100 + 87 + 50

= 237 yards

The expression that shows the distance that he walked = x + y + 100

= 87 + 50 + 100

= 237 yards

For more clarifications on trigonometry, visit: https://brainly.com/question/13729598

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