a wire long 10m long goes from the top of 7m pole to a point on a wall 12m above the ground. Find the angle between the wire and the wall?

To find the angle between the wire and the wall, we can use trigonometry.

First, let's draw a right-angled triangle to represent the situation:

|
|
| x (distance along the wall)
|
|----------------
| --------------
| /
| /
| /
| / 10m (length of the wire)
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|
|
(7m pole)

In this triangle, the hypotenuse is the length of the wire (10m), the side opposite the angle we want to find is the height of the wall (12m), and the side adjacent to the angle is the distance along the wall (x).

We can use the trigonometric function tangent (tan) to find the angle:

tan(angle) = opposite / adjacent

tan(angle) = 12 / x

To find the value of x, we can use the Pythagorean theorem:

x^2 + 7^2 = 10^2

x^2 + 49 = 100

x^2 = 100 - 49

x^2 = 51

x = sqrt(51)

Now we can substitute the value of x into the equation for tangent:

tan(angle) = 12 / sqrt(51)

To find the angle, we can take the inverse tangent (arctan) of both sides:

angle = arctan(12 / sqrt(51))

Using a calculator, the approximate value of the angle is 52.7 degrees.

Therefore, the angle between the wire and the wall is approximately 52.7 degrees.

To find the angle between the wire and the wall, we can use trigonometry. Let's break down the problem:

1. Draw a right-angled triangle representing the situation. The vertical side of the triangle represents the height of the pole (7m), the horizontal side represents the distance from the pole to the wall (10m), and the wire acts as the hypotenuse of the triangle.

|
7m |
|\
| \
--|__\-- 10m
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|
|
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|__ Wall (12m)

2. Use the Pythagorean theorem to find the length of the wire. According to the theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, let's call the length of the wire "w". We have:
w^2 = 7^2 + 10^2
= 49 + 100
= 149

Therefore, the length of the wire is w = √149 ≈ 12.2m.

3. Now that we have the lengths of the wire and the vertical side of the triangle, we can find the sine of the angle between the wire and the wall. The sine of an angle is equal to the ratio of the opposite side to the hypotenuse.

Let's call the angle θ (theta). We have:
sin(θ) = opposite / hypotenuse
= 12 / 12.2

Therefore, the sine of the angle is sin(θ) ≈ 0.9836.

4. To find the angle itself, we can use the inverse sine function (also known as arcsine or sin^(-1)) to calculate the angle when given the sine value.

θ ≈ sin^(-1)(0.9836) ≈ 80.2°

Therefore, the angle between the wire and the wall is approximately 80.2 degrees.

the wire is the hypotenuse of a right triangle

the difference between the top of the pole and the wire attachment is one side

cos(Θ) = (12 - 7) / 10