SOLUTION: In a town, Chief X resides 60m away on a bearing of 057° from the palace P, while Chief y resides on a bearing of 150° from the same palace P. The residence of X and Y are 180m a

Question 1202999: In a town, Chief X resides 60m away on a bearing of 057° from the palace P, while Chief y resides on a bearing of 150° from the same palace P. The residence of X and Y are 180m apart.
(a) Illustrate the information in a diagram.
Find, correct to three significant figures, the:
(i) bearing of X from Y;
(ii) distance between P and Y.
Answer by math_tutor2020(3821) (Show Source): You can put this solution on YOUR website!

Part (a)

This is one way to draw the diagram.

The triangle XPY is shown above with extra helper points A,B,C.

Point A is directly north of P.
Point A is useful to form the angles APX = 57 and APY = 150
These are the bearing angles.
The bearing angles have direct north as the starting direction, and then you turn clockwise toward the east when increasing the angles.

note:
(angle APX) + (angle XPY) = angle APY
57° + 93° = 150°

Point B is north of Y to help set up angle BYX. See the next section below.

Point C is south of P so we can form angle CPY, which will help determine angle CYP.

----------------------------------

For the next two parts, I'll give hints.

Part (i)

You'll need to...

Determine angle BYX
Subtract that angle from 360 to find the reflex angle, so you can determine the bearing of X from Y.

Note that
(angle CYP) + (angle PYX) + (angle BYX) = 90

Angle CYP is complementary to angle CPY = 30 shown in the diagram.
i.e.
(angle CPY) + (angle CYP) = 90
30 + (angle CYP) = 90
I'll let you finish this part.

Angle PYX is determined through the law of sines
Focus on triangle XPY to say:
sin(P)/XY = sin(Y)/PX
sin(93)/180 = sin(Y)/60
I'll let you finish this part.
This will determine angle PYX.

After determining angles CYP and PYX, you'll have enough info to determine angle BYX.

After finding angle BYX, subtract from 360.
The goal is to compute: 360 - (angle BYX)
so you can find the bearing of X from Y.

----------------------------------

To answer part (ii), you can use either the law of sines or law of cosines to find the distance from P to Y.

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