The 636 N weight held by two cables. The left-hand cable had tension 840 N and makes an angle of θ with the wall. The right-hand cable had tension 890 N and makes an angle of θ1 with the ceiling.
a) What is the angle θ1 which the right-hand cable makes with respect to the ceiling?
b) What is the angle θ which the left-hand cable makes with respect to the wall?
To find the angle θ1 which the right-hand cable makes with respect to the ceiling, we can use the law of cosines. The law of cosines states that in a triangle with side lengths a, b, and c, and angle θ opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab cos(θ)
In this case, the side lengths are the tensions in the cables, and the angle we are looking for is θ1.
Let's solve the equation for θ1:
890^2 = 840^2 + 636^2 - 2 * 840 * 636 * cos(θ1)
789210 = 705600 + 404496 - 1070658 * cos(θ1)
789210 - 1100096 = -1070658 * cos(θ1)
-311886 = -1070658 * cos(θ1)
Dividing both sides by -1070658:
cos(θ1) = -311886 / -1070658
cos(θ1) ≈ 0.291
To find the angle θ1, we can take the inverse cosine (arccos) of 0.291:
θ1 ≈ arccos(0.291)
Using a calculator, we find:
θ1 ≈ 74.7 degrees
Therefore, the angle θ1 which the right-hand cable makes with respect to the ceiling is approximately 74.7 degrees.
To find the angle θ which the left-hand cable makes with respect to the wall, we can use the same approach.
840^2 = 890^2 + 636^2 - 2 * 890 * 636 * cos(θ)
705600 = 792100 + 404496 - 1131240 * cos(θ)
705600 - 1194596 = -1131240 * cos(θ)
-489996 = -1131240 * cos(θ)
Dividing both sides by -1131240:
cos(θ) = -489996 / -1131240
cos(θ) ≈ 0.433
To find the angle θ, we can take the inverse cosine (arccos) of 0.433:
θ ≈ arccos(0.433)
Using a calculator, we find:
θ ≈ 64.8 degrees
Therefore, the angle θ which the left-hand cable makes with respect to the wall is approximately 64.8 degrees.
To find the angles θ and θ1, we can use trigonometric principles. Let's break down the problem step by step.
Let's start with part (a):
a) To find the angle θ1, which the right-hand cable makes with respect to the ceiling, we need to apply the trigonometric function called tangent (tan).
Let's consider the right-hand cable:
Tan(θ1) = Opposite / Adjacent
The opposite side is the tension in the right-hand cable (890 N), and the adjacent side is the weight being held by the cables (636 N). Therefore, we have:
Tan(θ1) = 890 N / 636 N.
To find θ1, we need to take the inverse tangent (also known as arctan or tan^(-1)) of both sides:
θ1 = arctan (890 N / 636 N).
Calculating this, we get the value of θ1.
Now, let's move on to part (b):
b) To find the angle θ, which the left-hand cable makes with respect to the wall, we can apply a similar process:
Tan(θ) = Opposite / Adjacent
In this case, the opposite side is the tension in the left-hand cable (840 N), and the adjacent side is the weight being held by the cables (636 N). Therefore, we have:
Tan(θ) = 840 N / 636 N.
To find θ, we need to take the inverse tangent of both sides:
θ = arctan (840 N / 636 N).
Calculating this will give us the value of θ.
So, by following these steps, we can determine the values of both θ and θ1.