Consider the 667 N weight held by two cables shown below. The left-hand cable had tension 890 N and makes an angle of è with the wall. The right-hand cable had tension 960 N and makes an angle of è1 with the ceiling.

a) What is the angle è1 which the right- hand cable makes with respect to the ceiling? Round up to four significant digits. Express unit as ”degrees”.

b) What is the angle è which the left-hand cable makes with respect to the wall? Round up to four significant digits. Express unit as ”degrees”.

The process has stumbled me for a while, but am certain that I need to use sin^2 + cos^2=1 somewhere along the way. I appreciate your help.

To determine the angles è and è1, we will use trigonometric principles and the given information. Let's start with angle è1.

a) To find angle è1, we can use the trigonometric function cosine (cos) since it relates the adjacent and hypotenuse sides of a right triangle. The right-hand cable makes an angle è1 with the ceiling, and we know the tension in that cable is 960 N.

Let's define the adjacent side and the hypotenuse side of the right triangle formed by the right-hand cable:
- The adjacent side is the tension in the cable (960 N).
- The hypotenuse side is the weight held by both cables (667 N).

Using the cosine function, we can write:

cos(è1) = adjacent / hypotenuse

cos(è1) = 960 N / 667 N

To find the angle è1, we need to take the inverse cosine (cos^-1) of both sides:

è1 = cos^-1(960 N / 667 N)

Now, let's solve this using a calculator:

è1 ≈ 17.4018 degrees (rounded to four significant digits)

Therefore, the angle è1 that the right-hand cable makes with respect to the ceiling is approximately 17.4018 degrees.

b) Similarly, to find the angle è, we can use the trigonometric function cosine (cos) since it relates the adjacent and hypotenuse sides of a right triangle. The left-hand cable makes an angle è with the wall, and we know the tension in that cable is 890 N.

Let's define the adjacent side and the hypotenuse side of the right triangle formed by the left-hand cable:
- The adjacent side is the tension in the cable (890 N).
- The hypotenuse side is the weight held by both cables (667 N).

Using the cosine function, we can write:

cos(è) = adjacent / hypotenuse

cos(è) = 890 N / 667 N

To find the angle è, we need to take the inverse cosine (cos^-1) of both sides:

è = cos^-1(890 N / 667 N)

Now, let's solve this using a calculator:

è ≈ 39.0466 degrees (rounded to four significant digits)

Therefore, the angle è that the left-hand cable makes with respect to the wall is approximately 39.0466 degrees.

In summary:
a) The angle è1 is approximately 17.4018 degrees.
b) The angle è is approximately 39.0466 degrees.