An Aircraft Flies Round A Triangular Curve,the First Leg Is 200km On A Bearing Of 115^o And The Second Leg Is 150km On A Bearing Of 230^o.How Long Is The Third Leg
To find the length of the third leg of the triangular curve, we need to use trigonometry and vector addition.
Let's break down the problem step by step:
Step 1: Draw the triangle
Start by sketching a diagram of the triangular curve described in the question. Label the three legs of the triangle as leg A, leg B, and leg C.
Step 2: Find the angle at point A
The bearing of 115° describes the angle at point A. Since the bearing is measured clockwise from the north, we need to subtract it from 360°.
Angle A = 360° - 115° = 245°
Step 3: Find the angle at point B
The bearing of 230° describes the angle at point B. Again, subtract it from 360°.
Angle B = 360° - 230° = 130°
Step 4: Find the angle at point C
To find the angle at point C, we subtract the sum of angles A and B from 180°.
Angle C = 180° - (Angle A + Angle B)
Step 5: Calculate the length of the third leg using the Law of Cosines.
The Law of Cosines states that for any triangle with sides a, b, and c, and the angle opposite side c, we can determine the length of side c using the following formula:
c² = a² + b² - 2ab * cos(C)
In this case:
Leg A = 200 km
Leg B = 150 km
c (the third leg) = ?
Let's plug in the values and calculate:
c² = (200)² + (150)² - 2(200)(150) * cos(C)
The value of cos(C) can be found by using the cosine rule to find the cosine of angle C.
cos(C) = cos⁻¹((a² + b² - c²) / (2ab))
cos(C) = cos⁻¹((200² + 150² - c²) / (2 * 200 * 150))
Once we've calculated cos(C), we substitute it back into the Law of Cosines equation to solve for c.
c² = (200)² + (150)² - 2(200)(150) * cos(C)
Finally, take the square root of c² to find the length of the third leg.
c = √(c²)
Simply plug in the values and follow the calculations to find the length of the third leg of the triangular curve.
find the angle between the headings (not bearings!)
use the law of cosines for the third side.