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. 3.0 yards 3.2 yards 3.8 yards 4.4 yards law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

Explanation:

Step1: Find angle ATB

In triangle ATB, the sum of interior angles is 180°. 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 $\frac{AB}{\sin\angle ATB}=\frac{AT}{\sin B}$. Given $AB = 1$ yard, $\angle ATB=77^{\circ}$, and $\angle B = 46^{\circ}$. So, $AT=\frac{AB\times\sin B}{\sin\angle ATB}=\frac{1\times\sin46^{\circ}}{\sin77^{\circ}}$. 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, $\sin A=\frac{h}{AT}$. Since $\angle A = 57^{\circ}$ and $AT\approx0.7382$ yards, then $h = AT\times\sin A$. $\sin57^{\circ}\approx0.8387$, so $h=0.7382\times0.8387\approx3.2$ yards.

Answer:

3.2 yards