Sine theorem
We are going to use in this problem the Law of Sines:
[tex]\frac{A}{\sin(a)}=\frac{B}{\sin (b)}[/tex]This is, in any triangle if we divide a side with its opposite angle sine,we will obtain the same result.
Sine theorem applied to this case
In this case:
the side h and the angle tº are opposite
and
the side y and the angle ∠LGM are opposite
then,
[tex]\frac{h}{\sin(tº)}=\frac{y}{\sin (\angle LGM)}[/tex]Finding the angle ∠LGM
Using the given information, we have that the angles sº and tº are complementary, this means that their together form a 90º angle:
sº + tº = 90º
then
sº = 90º - tº
Since the sum of all the inner angles of a triangle is 180º, and we have that ΔLGM is a right triangle, then
∠LGM + tº + 90º = 180º
then for ∠LGM, we have that
∠LGM = 180º - 90º - tº
∠LGM = 90º - tº
then ∠LGM and sº are the same angle: 90º - tº
∠LGM = sº
We can replace sº by ∠LGM in the equation we found:
[tex]\begin{gathered} \frac{h}{\sin(tº)}=\frac{y}{\sin(\angle LGM)} \\ \downarrow \\ \frac{h}{\sin(tº)}=\frac{y}{\sin(sº)} \end{gathered}[/tex]If we multiply both sides by sin(tº), we have:
[tex]\begin{gathered} \frac{h}{\sin(tº)}=\frac{y}{\sin(sº)} \\ \downarrow \\ h=\frac{y}{\sin(sº)}\cdot\sin (tº) \end{gathered}[/tex]Answer A: h = y · sin(tº)/sin(sº)
We have that y = 3m, and sº = 38º
in order to find h using the equation we found, we must find tº. Since sº and tº are complementary,
sº + tº = 90º
then
tº = 52º
Now, we can replace in the equation:
[tex]\begin{gathered} h=\frac{y}{\sin(sº)}\cdot\sin (tº) \\ \downarrow \\ h=\frac{3m}{\sin (38º)}\cdot\sin (52º) \end{gathered}[/tex]since
sin(52º) ≅ 0.79
and
sin (38º) ≅ 0.62
then
[tex]\begin{gathered} h=\frac{3m}{0.62}\cdot0.79 \\ \downarrow \\ h=3m\cdot1.27=3.82m \end{gathered}[/tex]Then, the height of the tree would be 3.82m
Answer B: 3.82 meters
