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 at the top of the triangle

The sum of angles in a triangle is $180^{\circ}$. The angles at the base are $57^{\circ}$ and $46^{\circ}$, so the angle at the top $T = 180-(57 + 46)=77^{\circ}$.

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

In $\triangle ABT$, by the law of sines $\frac{AB}{\sin T}=\frac{AT}{\sin B}$. Given $AB = 1$ yard, $T=77^{\circ}$, and $B = 46^{\circ}$. So $AT=\frac{AB\times\sin B}{\sin T}=\frac{1\times\sin46^{\circ}}{\sin77^{\circ}}$. 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 $A = 57^{\circ}$ and $AT\approx0.7382$ yards, then $h = AT\times\sin A$. $\sin57^{\circ}\approx0.8387$, so $h=0.7382\times0.8387\approx 3.0$ yards.

Answer:

3.0 yards