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. Two major considerations to keep in mind when working out these problems are: which trig function you should use, and what the units or measures are in the answer.
The missing values in the ratios of the trig functions represent the missing parts in the problems. You assign the known values appropriately and solve for what’s left.
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? The preceding figure shows the situation in pictorial form.
Identify the parts of the right triangle that you can use to solve the problem.
The knight assumes that the tower is perpendicular to the ground, forming a right triangle. He knows 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 he is standing) to the base of the tower is 15 feet. The hypotenuse is the length needed for the ladder — call it x. The following figure shows you the triangle.The right triangle that will help save the damsel in distress.
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 input the values that you know.
Solve the equation.
Cross-multiplying, you get
The ladder needs to be 30 feet long. (That knight had better be pretty strong!)