Pqr is allocated in the same horizontal plain, the bearing of q from p is130° and the bearing of r from q is 075°, then pq=6cm and qr =4cm, find the bearing of r from p, collect to the nearest degree
to use law of cosines we need angle PQR
PQ is 40deg down from x direction (East)
QR is 15 deg up from x direction (East)
so angle PQR is 180 - 40 - 14 = 126 deg
then
q^2 = p^2 + r^2 - 2 p r cos 126
q^2 = 4^2 + 6^2 - 2*4*6 cos 126
solve for q
then get the other angles from law of sines
eg:
sinQPR / 4 = sin 126 / q
All angles are measured CW from +y-axis,
PR = PQ + QR = 6cm[130o] + 4cm[75o],
X = 6*sin130 + 4*sin75 = 8.46 cm,
Y = 6*Cos130 + 4*Cos75 = -2.82 cm,
TanA = X/Y,
A = -71.6o = 71.6o E. of S. = 108o,
The bearing of R from P = 108o.
To find the bearing of point r from point p, we can use the concept of relative bearings.
Let's first draw a diagram to represent the given information:
```
P -------> Q -------> R
130° 75°
6cm 4cm
```
Now, to find the bearing of point R from point P, we need to find the angle between the line segments PQ and QR.
We can use the fact that the sum of interior angles in a triangle is 180°. In triangle PQR, angle P + angle Q + angle R = 180°.
We already know that angle Q is 130° and angle R is 75°.
So, angle P = 180° - angle Q - angle R
= 180° - 130° - 75°
= 180° - 205°
= -25°
Since the bearing is measured clockwise from the north, we need to convert -25° to its equivalent positive bearing:
Bearing of R from P = 360° - |-25°|
= 360° - 25°
= 335° (to the nearest degree)
Therefore, the bearing of point R from point P is 335° (to the nearest degree).
To find the bearing of point R from point P, we can use the concept of relative bearings.
First, let's understand the given information. We have a triangle PQR, where P and Q are given bearings, and the lengths PQ = 6 cm and QR = 4 cm.
Step 1: Draw the diagram
Start by drawing a straight line segment representing PQ with a length of 6 cm. From point P, draw a line segment at a bearing of 130°, which represents the bearing of Q from P. Then, from point Q, draw a line segment at a bearing of 75°, which represents the bearing of R from Q.
P -------6 cm--------> Q---------------------> R
Step 2: Find the bearing of R from P
To find the bearing of R from P, we need to determine the angle PRQ (θ), since the bearing of R from P will be the complement of θ.
Using the law of cosines, we can calculate the angle PRQ:
PRQ = arccos((PQ^2 + QR^2 - RP^2) / (2 * PQ * QR))
Substituting the given values:
PRQ = arccos((6^2 + 4^2 - RP^2) / (2 * 6 * 4))
Step 3: Solve for RP
To solve for RP, we can use the law of cosines again:
RP^2 = PQ^2 + QR^2 - 2 * PQ * QR * cos(PRQ)
Substituting the given values:
RP^2 = 6^2 + 4^2 - 2 * 6 * 4 * cos(PRQ)
Step 4: Find the value of PRQ
Now, substitute the value of RP^2 from Step 3 into the equation in Step 2 and solve for PRQ:
PRQ = arccos((6^2 + 4^2 - 2 * 6 * 4 * cos(PRQ)) / (2 * 6 * 4))
Using the inverse cosine function on your calculator, find the value of PRQ.
Step 5: Calculate the bearing of R from P
The bearing of R from P is the complement of PRQ:
Bearing of R from P = 180° - PRQ
Calculate this value and round it to the nearest degree.
This will give you the bearing of point R from point P.