To determine the distance of point [tex]\( P \)[/tex] to the north of point [tex]\( Q \)[/tex], given that the bearing from [tex]\( Q \)[/tex] to [tex]\( P \)[/tex] is [tex]\( 61^\circ \)[/tex], and the distance between [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] is [tex]\( 40 \)[/tex] km, we can use trigonometry. Here is a detailed, step-by-step solution:
1. Understand Bearing and Distance:
- A bearing of [tex]\( 61^\circ \)[/tex] means that the angle measured clockwise from the north direction to the line connecting [tex]\( Q \)[/tex] to [tex]\( P \)[/tex] is [tex]\( 61^\circ \)[/tex].
- The given distance between [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] is [tex]\( 40 \)[/tex] km.
2. Visualize the Situation:
- Let's imagine a right triangle where:
- Point [tex]\( Q \)[/tex] is at the origin.
- Point [tex]\( P \)[/tex] is [tex]\( 40 \)[/tex] km away from [tex]\( Q \)[/tex] at a bearing of [tex]\( 61^\circ \)[/tex].
3. Identify the Required Component:
- Since we need to find how far [tex]\( P \)[/tex] is to the north of [tex]\( Q \)[/tex], we need the vertical component of this distance in the direction that points north.
4. Use Trigonometry:
- In a right triangle, the vertical component (north distance) can be found using the cosine of the angle when the hypotenuse and angle are known.
- Cosine relates the adjacent side of a right triangle to the hypotenuse. Here, the adjacent side corresponds to the northward distance we are looking for.
5. Apply the Cosine Function:
- The northward distance, let's call it [tex]\( d \)[/tex], is given by:
[tex]\[
d = 40 \cos(61^\circ)
\][/tex]
6. Calculate the Value:
- Using the cosine function and the specified angle, we find:
[tex]\[
d = 40 \cos(61^\circ)
\][/tex]
7. Result:
- The calculation yields a northward distance of approximately [tex]\( 19.39 \)[/tex] km.
Therefore, the distance of point [tex]\( P \)[/tex] to the north of point [tex]\( Q \)[/tex] is [tex]\( 19.39 \)[/tex] km.
