A man is standing on a hill 200 meters high. If he looks down at an angle of dep

A man is standing on a hill 200 meters high. If he looks down at an angle of depression of 30 degrees, how far is he from the base of the hill?

A man is standing on a hill 200 meters high. If he looks down at an angle of depression of 30 degrees, how far is he from the base of the hill?

Step 1: Understand the problem. A man is on a hill that is 200 meters high.
Step 2: The angle of depression is the angle between the horizontal line from the man and the line of sight down to the base of the hill. Here, it is 30 degrees.
Step 3: Visualize a right triangle where the height of the hill (200 meters) is the vertical side, and the distance from the man to the base of the hill is the horizontal side.
Step 4: The angle of depression (30 degrees) is equal to the angle of elevation from the base of the hill to the man.
Step 5: Use the tangent function, which relates the opposite side (height of the hill) to the adjacent side (distance from the base). The formula is: tan(angle) = opposite / adjacent.
Step 6: Rearrange the formula to find the distance (adjacent side): distance = height / tan(angle).
Step 7: Substitute the values into the formula: distance = 200 / tan(30 degrees).
Step 8: Know that tan(30 degrees) = 1/√3.
Step 9: Substitute tan(30 degrees) into the formula: distance = 200 / (1/√3).
Step 10: Simplify the equation: distance = 200 * √3.
Step 11: Calculate the final distance: distance = 200√3 meters.

Trigonometry – The problem involves using the tangent function to relate the height of the hill to the distance from the base, given an angle of depression.
Angle of Depression – Understanding that the angle of depression from the man's line of sight to the base of the hill is equal to the angle of elevation from the base to the man.