Question

two students stand 1 yard apart and measure their respective angles of elevation to the top of a tree. student a measures the angle to be 57°, and student b measures the angle to be 46°. what is h, the height of the tree? use the law of sines to first find at. then use that measure to find the value of h. law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

Explanation:

Step1: Find angle $\angle ATB$

In $\triangle ATB$, the sum of interior - angles of a triangle is $180^{\circ}$. Given $\angle TAB = 57^{\circ}$ and $\angle TBA=46^{\circ}$, so $\angle ATB=180^{\circ}-(57^{\circ}+46^{\circ}) = 77^{\circ}$.

Step2: Use the law of sines to find $AT$

In $\triangle ATB$, by the law of sines $\frac{AB}{\sin\angle ATB}=\frac{AT}{\sin\angle TBA}$. We know $AB = 1$ yard, $\angle ATB = 77^{\circ}$, and $\angle TBA = 46^{\circ}$. So $AT=\frac{AB\times\sin\angle TBA}{\sin\angle ATB}=\frac{1\times\sin46^{\circ}}{\sin77^{\circ}}$. Since $\sin46^{\circ}\approx0.7193$ and $\sin77^{\circ}\approx0.9744$, then $AT=\frac{0.7193}{0.9744}\approx0.7382$ yards.

Step3: Find the height $h$ of the tree

In right - triangle $AGT$, $\sin\angle TAG=\frac{h}{AT}$. Since $\angle TAG = 57^{\circ}$ and $AT\approx0.7382$ yards, then $h = AT\times\sin57^{\circ}$. Since $\sin57^{\circ}\approx0.8387$, $h=0.7382\times0.8387\approx3.2$ yards.

Answer:

3.2 yards