Use trigonometry to help you solve the following. A person is standing halfway between two trees that are 3m apart. The angles of elevation to the tops of the trees are 42^circ and 36^circ How much taller is one tree over the other? The one tree is square taller than the other.

To solve this problem, we can use trigonometry. Let's label the trees as Tree A and Tree B, with Tree A being the taller tree.<br /><br />Step 1: Draw a diagram to represent the situation. Place the person halfway between the two trees, with the angle of elevation to the top of Tree A being $42^{\circ}$ and the angle of elevation to the top of Tree B being $36^{\circ}$.<br /><br />Step 2: Use the tangent function to find the height of each tree. The tangent of an angle in a right triangle is equal to the opposite side divided by the adjacent side.<br /><br />For Tree A:<br />$\tan(42^{\circ}) = \frac{{\text{{height of Tree A}}}}{{\text{{distance from person to Tree A}}}}$<br /><br />For Tree B:<br />$\tan(36^{\circ}) = \frac{{\text{{height of Tree B}}}}{{\text{{distance from person to Tree B}}}}$<br /><br />Step 3: Since the person is standing halfway between the two trees, the distance from the person to each tree is the same. Let's call this distance $d$.<br /><br />For Tree A:<br />$\tan(42^{\circ}) = \frac{{\text{{height of Tree A}}}}{d}$<br /><br />For Tree B:<br />$\tan(36^{\circ}) = \frac{{\text{{height of Tree B}}}}{d}$<br /><br />Step 4: Solve for the height of each tree.<br /><br />Height of Tree A = $d \cdot \tan(42^{\circ})$<br /><br />Height of Tree B = $d \cdot \tan(36^{\circ})$<br /><br />Step 5: Calculate the difference in height between the two trees.<br /><br />Difference in height = Height of Tree A - Height of Tree B<br /><br />Therefore, the one tree is $\square$ taller than the other.