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 solve this problem, you can use the equations involving trigonometric functions and the concept of equilibrium.

First, let's analyze the forces acting on the weight:

1. The weight itself, which is 667 N, acting downward.

2. The tension in the left-hand cable, which is 890 N, acting at an angle è with the wall.

3. The tension in the right-hand cable, which is 960 N, acting at an angle è1 with the ceiling.

Now, let's break down these forces into their vertical and horizontal components.

For the weight:

- The vertical component is equal to the weight itself, which is 667 N.

- The horizontal component is zero since the weight is acting vertically.

For the left-hand cable:

- The vertical component is equal to the tension in the left-hand cable, which is 890 N, multiplied by the sine of è.

- The horizontal component is equal to the tension in the left-hand cable, which is 890 N, multiplied by the cosine of è.

For the right-hand cable:

- The vertical component is equal to the tension in the right-hand cable, which is 960 N, multiplied by the sine of è1.

- The horizontal component is equal to the tension in the right-hand cable, which is 960 N, multiplied by the cosine of è1.

Since the weight is in equilibrium, the sum of the vertical components of all the forces must be zero, and the sum of the horizontal components of all the forces must also be zero.

Using these conditions, we can set up the following equations:

Vertical equilibrium equation:
667 N + 890 N * sin(è) + 960 N * sin(è1) = 0

Horizontal equilibrium equation:
890 N * cos(è) + 960 N * cos(è1) = 0

Now, to solve for angle è1:

a) Rearrange the vertical equilibrium equation to solve for sin(è1):
sin(è1) = -(667 N + 890 N * sin(è)) / 960 N

b) Use the arcsine function to get the angle è1 in radians:
è1 = arcsin(-(667 N + 890 N * sin(è)) / 960 N)

c) Convert the angle to degrees by multiplying by 180/π:
è1_degrees = è1 * 180/π

d) Round the result to four significant digits.

Now, to solve for angle è:

a) Plug the value of è1 obtained in the previous step into the horizontal equilibrium equation:
890 N * cos(è) + 960 N * cos(è1) = 0

b) Solve this equation numerically or algebraically to find the value of angle è.

c) Convert the angle to degrees by multiplying by 180/π.

d) Round the result to four significant digits.

Note: The equation sin^2(è) + cos^2(è) = 1 is the Pythagorean identity and is used to relate the sine and cosine of an angle in a right triangle. However, in this problem, we are dealing with forces in a two-dimensional system, rather than a right triangle, so the equation is not directly applicable.