I know I posted this before,but I did not learn it your way .....instead I drew a picture and use the law of cosines....is there a way you can show me the work using law of cosines??
A plane flies 30 mi on a bearing of 200 degrees and then turns and flies 40 mi on a bearing of 10 degrees. Find the resultant displacement vector as a distance and bearing.
sure,
After sketching in all the angles, and doing some arithmetic, I have a triangle with sides
30 and 40 and the angle between them as 10°
let the resultant be R
R^2 = 30^2 + 40^2 - 2(30)(40)cos10°
= 900+1600-2400cos10°
= 136.46..
R = √136.46
= 11.68 , which I believe I had before
for the angle, let Ø be the obtuse angle
sinØ/40 = sin10°/11.68
sinØ = 40sin10°/11.68 = .5947
Ø = 36.5° or 143.5°
but it is obviously obtuse,
so the bearing is 143.5 + 200 = 343.5°
I can't recall what I had in the other solution
Well if you add the sin and cosine you form an equilateral triangle, then you take the area of the triangle and form a helix,and the area of that 4. then you take the 4 cosine to that and add it to adjacent side. answer is 5
Of course! I can show you how to use the law of cosines to find the resultant displacement vector. The law of cosines relates the sides and angles of a triangle. In this case, we can consider the triangle formed by the initial position, the final position, and the origin (a point of reference).
To find the resultant displacement vector, we need to find the magnitude (distance) and the direction (bearing) of the vector. Let's start by finding the magnitude using the law of cosines.
Step 1: Label the given information:
- First leg of the triangle: 30 mi
- Second leg of the triangle: 40 mi
- Included angle between the legs: 180 degrees - (bearing of 200 degrees + bearing of 10 degrees) = 340 degrees
Step 2: Apply the law of cosines:
The law of cosines states that:
c^2 = a^2 + b^2 - 2ab * cos(C)
Here, c represents the side opposite to the included angle C, and a and b represent the other two sides of the triangle.
So, applying the law of cosines, we have:
c^2 = (30 mi)^2 + (40 mi)^2 - 2 * 30 mi * 40 mi * cos(340 degrees)
Step 3: Calculate the magnitude:
c = sqrt((30 mi)^2 + (40 mi)^2 - 2 * 30 mi * 40 mi * cos(340 degrees))
After substituting the values and evaluating this equation, you will find the magnitude of the resultant displacement vector.
Now, to find the bearing or direction, we can use the concept of vectors and trigonometry.
Step 4: Calculate the bearing:
To find the bearing, we can consider the triangle formed by the initial position, final position, and north direction.
We can use the angle between the initial position, final position, and the north direction, which is equal to the complement of the bearing of 200 degrees (<BNP) minus the bearing of 10 degrees (<PNQ).
Step 5: Use trigonometry:
We can use the trigonometric functions tangent or sine to find the angle <NPQ, and then subtract it from 180 degrees to obtain the bearing.
Using the tangent function:
tan(<NPQ) = opposite/adjacent = 40 mi / 30 mi
<NPQ = arctan(40 mi / 30 mi)
Subtracting from 180 degrees will give you the bearing.
Please note that for trigonometric functions, make sure your calculator is set to the correct angle mode, either degrees or radians, as specified in the problem.
By following these steps, you can find the resultant displacement vector using the law of cosines and determine its magnitude and bearing.