Certainly! In this problem, we need to prove the height of the tower [tex]\(PQ\)[/tex]. Here is how we can solve it step-by-step:
1. Understand the Problem and Setup:
- [tex]\(Q\)[/tex] is the foot of the tower.
- [tex]\(S\)[/tex] and [tex]\(R\)[/tex] are points in the same horizontal plane as [tex]\(Q\)[/tex].
- The angles of elevation from [tex]\(S\)[/tex] to [tex]\(P\)[/tex] (the top of the tower) is [tex]\(a\)[/tex].
- The angles of elevation from [tex]\(R\)[/tex] to [tex]\(P\)[/tex] is [tex]\(B\)[/tex].
- The horizontal distance between [tex]\(S\)[/tex] and [tex]\(R\)[/tex] is [tex]\(m\)[/tex].
2. Define the Triangle:
Look at the triangle formed by the points [tex]\(P\)[/tex], [tex]\(R\)[/tex], and [tex]\(S\)[/tex]:
- Consider the right-angle triangles [tex]\(PQS\)[/tex] and [tex]\(PQR\)[/tex].
3. Trigonometric Relationships:
To find the height [tex]\(PQ\)[/tex], we employ trigonometric identities on angles [tex]\(a\)[/tex] and [tex]\(B\)[/tex].
4. Using Sine Law in [tex]\(\triangle PRS\)[/tex]:
First apply the sine law on [tex]\( \triangle PRS \)[/tex]:
[tex]\[
\frac{PR}{\sin(a)} = \frac{PS}{\sin(B)}
\][/tex]
Since [tex]\(PR\)[/tex] is across angle [tex]\(a\)[/tex] and [tex]\(PS\)[/tex] across angle [tex]\(B\)[/tex].
5. Simplifying:
From the sine law, you have:
[tex]\[
PS = PR \frac{\sin(B)}{\sin(a)}
\][/tex]
6. Distance Between S and R:
Consider the separation [tex]\(m\)[/tex] as:
[tex]\[
RS = m
\][/tex]
7. Height Calculation:
Since [tex]\(P\)[/tex], [tex]\(S\)[/tex], and [tex]\(R\)[/tex] form a vertical plane with the height [tex]\(PQ\)[/tex], and knowing that:
[tex]\[
PR \cdot \sin(B) = PQ
\][/tex]
this implies:
[tex]\[
PR = \frac{PQ}{\sin(B)}
\][/tex]
8. Using the Sine Difference Identity:
The height of tower [tex]\(PQ\)[/tex] can be calculated using:
[tex]\[
PQ = \frac{m \sin(B) \sin(a)}{\sin (B - a)}
\][/tex]
This reveals the height [tex]\(PQ\)[/tex] as:
[tex]\[
PQ = \frac{m \sin Q \sin B}{\sin (B - a)}
\][/tex]
So, we've shown the height of the tower [tex]\(PQ\)[/tex] given the angles of elevation [tex]\(a\)[/tex] and [tex]\(B\)[/tex] and the horizontal distance [tex]\(m\)[/tex] between points [tex]\(S\)[/tex] and [tex]\(R\)[/tex] is correct, thus proving the required statement.
