Three towns pqr are such that the distance between p and q is 90km.if the bearing of q is 075 and the bearing of R from P is 310 find the
A.distance between Q and R
B. Bearing of R from Q
in triangle QPR, angle QPR is 75+50 = 125°
All we have so far is one side (PQ=90) and one angle.
That does not determine a triangle.
Figure out what is missing, then use the law of sines or law of cosines as needed.
A. QR/sin P = PQ/sin R.
QR/sin125 = 90/sin40
QR = 115 km.
B. PR/sin Q = PQ/sin R.
PR/sin15 = 90/sin40
PR = 36 km.
QR = QP+PR
QR = 90[255] + 36[310]
QR = (90*sin255+36*sin310) + (90*cos255+36*cos310)i
QR = -115 - 0.15i
Tan A = -115/-0.15
A = 89.9o CW.
To find the distance between Q and R, we can use the concept of the Sine Rule. The Sine Rule states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. Using this rule, we can set up the following equation:
sin(Q) / QR = sin(P) / PR
In this case, we are given the bearing of Q, which is 075. Since bearing is measured clockwise from north, we need to convert it to an angle in trigonometric terms. We can do this by subtracting the given bearing from 090 (which represents north). Therefore, the angle Q is 090 - 075 = 015 degrees.
Similarly, we are given the bearing of R from P, which is 310. Subtracting this bearing from 360 degrees gives us the angle P, which is 360 - 310 = 050 degrees.
Now, let's substitute these values into the Sine Rule equation:
sin(015) / QR = sin(050) / PR
Next, we can rearrange the equation to solve for QR:
QR = (sin(015) * PR) / sin(050)
However, we don't yet know the length of PR. To find this, we can use the Cosine Rule, which states that for any triangle:
c² = a² + b² - 2ab * cos(C)
In this case, we can use the Cosine Rule to find PR. Rearranging the equation to solve for PR:
PR² = PQ² + QR² - 2 * PQ * QR * cos(P)
Given that PQ is 90 km, we can substitute the values and solve for PR:
PR² = 90² + QR² - 2 * 90 * QR * cos(050)
Now, let's solve this quadratic equation for QR. Once we have the value of QR, we can substitute it back into the Sine Rule equation to find the value of PR.