a Recall the definition of the cosine of an angle.
B
b Recall the definition of the tangent of an angle.
A
a See solution.
B
b See solution.
Practice makes perfect
a We are given that a communications tower is located on a plot of flat land and is supported by several guy wires. Let's take a look at the given diagram.
We are asked to explain the way to estimate the length of any guy wire, let's call it l, assuming that we are able to measure distances along the ground d and the angles formed by the guy wires and the ground α. Let's focus on one guy wire.
Now let's recall that the cosine of an angle is the ratio between the leg adjacent to this angle and the hypotenuse. Therefore, if we assume that the tower is perpendicular to the ground, we can write an equation for cos α.
cos α=d/l
Finally, we will solve the above equation for l using inverse operations.
Therefore, we can estimate the length of any guy wire by dividing its distance along the ground by the cosine of an angle it forms with the ground.
b This time we are asked to estimate how high on the tower each wire is attached. Let's call this height h. Again, we will assume that we are able to measure distances along the ground d and the angles formed by the guy wires and the ground α. Let's focus on one guy wire like we did before.
Now let's recall that the tangent of an angle is the ratio between the leg opposite this angle and the leg adjacent to this angle. Therefore, if we assume that the tower is perpendicular to the ground, we can write an equation for tan α.
tan α=h/d
Finally, we will solve the above equation for h using inverse operations.
Therefore, we can estimate how high on the tower each wire is attached by multiplying its distance along the ground and the tangent of an angle it forms with the ground.
