A 60-foot tree casts a shadow 85 feet long. The sine of the angle of elevation of the top of the tree to the sun is approximately ___.

A. .82
B. .75
C. .58
D. .33

.58

To find the sine of the angle of elevation, we need to use the relationship between the length of the shadow and the height of the tree.

The sine of an angle is equal to the opposite side divided by the hypotenuse.

In this case, the height of the tree is the opposite side, and the length of the shadow is the hypotenuse.

Let's solve for the sine of the angle of elevation:

sin(angle) = height / shadow

sin(angle) = 60 / 85

Calculating this gives us:

sin(angle) ≈ 0.70588235294

Rounding this to two decimal places, the approximate value of the sine of the angle of elevation is 0.71.

None of the given choices match exactly, so the closest option is:

B. 0.75

To find the sine of the angle of elevation of the top of the tree to the sun, we can use the information given about the tree's height and shadow length.

The angle of elevation is the angle by which the sun's rays rise above the horizon. We can set up a right triangle where the height of the tree is the side opposite to the angle of elevation, and the shadow length is the side adjacent to the angle of elevation.

Let's denote the angle of elevation as θ. To find θ, we can use the inverse tangent function:

θ = arctan(opposite/adjacent)

In this case, the opposite side is the height of the tree, which is 60 feet, and the adjacent side is the shadow length, which is 85 feet. Therefore, we have:

θ = arctan(60/85)

Using a calculator or trigonometric tables, we find that θ is approximately 36.87 degrees.

Next, we need to find the sine of θ. Sine is defined as the ratio of the opposite side to the hypotenuse in a right triangle. The hypotenuse in our case is the distance from the top of the tree to the sun, which we don't know.

However, we can use the fact that the shadow of the tree forms a similar triangle with the tree itself and its shadow. The ratio of corresponding sides in similar triangles is the same, so:

tree height / shadow length = distance to the sun / tree shadow length

Plugging in the given values, we have:

60 / 85 = distance to the sun / 85

Simplifying, we find that the distance to the sun is 60 feet.

Now, we can find the sine of θ:

sine(θ) = opposite/hypotenuse = 60/60 = 1

Therefore, the sine of the angle of elevation of the top of the tree to the sun is 1.

None of the given answer choices match with our calculation.

it's exactly 60/√(60^2+85^2)