Using Trigonometry to See if a Ladder Reaches a Window
Every day, people use trigonometry to measure things that they can’t reach. How high is that building? Will this ladder reach to the top of that tree? By using the appropriate trig functions, you can find answers to such questions.
Consider the oh-so-common scenario: A damsel is in distress and is being held captive in a tower. Her knight in shining armor is on the ground below with a ladder. He needs to know whether it’ll reach her or whether he needs a longer ladder.
When the stunning knight stands 15 feet from the base of the tower and looks up at his precious damsel, the angle of elevation to her window is 60 degrees. How long does the ladder have to be?
Identify the parts of the right triangle that you can use to solve the problem.
You know that the acute angle is 60 degrees, and the adjacent side of the triangle is along the ground; the distance from the vertex of the angle (where the knight is standing) to the base of the tower is 15 feet (the adjacent side). The hypotenuse is the length needed for the ladder — call it x.
Determine which trig function to use.
The adjacent side and hypotenuse are parts of the cosine ratio. Those sides are also parts of the secant ratio, but if at all possible, you should use the three main functions, not their reciprocals.
Write an equation with the trig function; then insert the values that you know.
The cosine of 60 degrees is 1/2, the adjacent side is 15 feet, and the hypotenuse is unknown.
Solve the equation.
Cross-multiplying, you get
The ladder needs to be 30 feet long. (That knight had better be pretty strong!)