From the top of a lighthouse 180 feet high, the angle of depression of a boat is 23o. find the distance from the boat to the foot of the lighthouse to the nearest foot. (the lighthouse was built at sea level.)

The lighthouse is 424 feet away.

Step-by-step explanation:

There is no other way to do this but to use one of the 6 trigonometry functions.

Drawing

Draw a dotted horizontal line.

judge an angle that could be 23o. Let it slant downward from the left side of the dotted line.

Draw another horizontal line that represents sea level.

Join the left side of the dotted line to the last line you drew. The angle on your right is also 23o.

Function

You have 6 trig functions to choose from. You have the lighthouse height (180 feet) the angle on your right, and the length on the horizontal representing the distance from the lighthouse base to the boat.

You have an angle

You have an opposite side

You have a horizontal line (the adjacent side)

You want to use the tangent function.

Tan(23) = opposite / ad

Tan(23) = 180 / adjacent     Multiply both sides by the adjacent

adjacent*Tan(23)= 180        Divide by Tan(23)

adjacent = 180/Tan(23)

adjacent = 180 / 0.42447

adjacent = 424 feet.

the perimeter is increased by a factor of 2