A road start at college and goes due North for 2000m it then 2000m on a bearing of 040 degrees and end it a market how far is the market from the college and what is the bearing of the market from the college
All angles are measured CW from +y-axis.
D = 2km[0o]+2km[40o].
D = (2*sin0+2*sin40)+(2*cos0+2*cos40)i
D = 1.29+3.53i = 3.76km[20o] = 3760m[20o].
I scaled it down by a factor of 1000
and got a triangle of sides 2 with an angle of 140° between them
If x is the distance we want,
x^2 = 2^2 + 2^2 - 2(2)(2)cos140
x = 3.75877..
so the distance between them is appr 3759 metres
angle:
sinØ/2 = sin140/3.75877
sincØ = .....
Ø = .....
let me know what you get for Ø
Do you realize you could have gotten Ø without the sine law?
I don't understand please 🥺 explain it well not only the answers
Sure, let me try to explain it step by step.
The problem gives us two legs of a triangle and an angle between those two legs. We want to find the length of the third side (the distance between the college and the market) and the angle between the first leg (the road going due North) and the third side.
First, we can use the cosine law to find the length of the third side:
c^2 = a^2 + b^2 - 2ab cos(C)
Here, a = b = 2 km (since we scaled everything down by a factor of 1000), and C = 140 degrees (since that's the angle between the two legs). Plugging in the values, we get:
c^2 = 2^2 + 2^2 - 2(2)(2)cos(140)
c^2 ≈ 14.119
c ≈ 3.75877 km
So the distance between the college and the market is about 3.75877 km, or 3758.77 meters (rounded to five decimal places).
Next, we want to find the angle between the first leg and the third side. To do this, we can use the sine law:
sin(A) / a = sin(B) / b = sin(C) / c
Here, we want to find A, the angle between the first leg and the third side. We know c (the length of the third side) and we know B (the angle between the two legs, which we already used to find c). So we just need to find the value of sin(A) and we can solve for A.
Using the formula above, we get:
sin(A) / 2 = sin(140) / 3.75877
sin(A) ≈ 0.67051
A ≈ 41.222 degrees
So the angle between the first leg and the third side is about 41.222 degrees (rounded to three decimal places).
Finally, we can check our work by adding up the three angles of the triangle to make sure they add up to 180 degrees:
Angle A = 41.222 degrees (already calculated)
Angle B = 140 degrees (given in the problem)
Angle C = 180 - A - B ≈ 18.778 degrees
So our work checks out!
Huuu
I got confused