A ladder eight meters long leans
against the wall. Find the horizontal length between
the base of the ladder and the wall, given the angle between the ladder and the
ground is 45 degrees.
We’ll begin this question by
drawing a diagram to include the ladder, the wall, and the ground. These three lines form a
right-angled triangle as the ground is horizontal and the wall vertical. We’re told that the ladder is eight
meters long and the angle between the ladder and the ground is 45 degrees. We’re asked to find the horizontal
length between the base of the ladder and the wall, which we’ll refer to as 𝑥
So the setup in this question is we
have a right-angled triangle in which we know the length of one side, we know the
size of one angle, and we want to calculate the length of the second side. This is the perfect setup to use
trigonometry. We’ll begin by labelling the three
sides of the triangle in relation to the angle of 45 degrees.
The ladder forms the hypotenuse of
this triangle, the wall forms the opposite as it’s opposite the angle of 45 degrees,
and the ground forms the adjacent side as it’s between the angle of 45 degrees and
the right angle. The two sides that are going to be
involved in the trigonometric ratio in this question are the adjacent 𝑥 metres and
the hypotenuse eight meters.
To identify which of the three
trigonometric ratios we need in this question, we can recall the acronyms SOHCAHTOA
to help, where S, C, and T stand for sine, cosine, and tangent and O, A, and H stand
for opposite, adjacent, and hypotenuse. A and H appear together in the CAH
part of this acronym. And therefore, it’s the cosine
ratio that we need in this question. Let’s recall its definition.
The cosine or cos of an angle 𝜃 is
equal to the adjacent divided by the hypotenuse. In this question, the angle 𝜃 is
45 degrees, the adjacent is 𝑥 metres, and the hypotenuse is eight meters. So we have the equation cos of 45
degrees is equal to 𝑥 over eight. In order to find the value of 𝑥,
we need to solve this equation. As this is coming in eight in the
denominator of our fraction, let’s multiply both sides of the equation by eight. This gives an expression for 𝑥: 𝑥
is equal to eight multiplied by cos of 45 degrees.
We need to evaluate this. Now, when using trigonometry to
answer a question where the angle involved is 45 degrees, it’s usual that you
wouldn’t have access to a calculator. The reason for this is that 45
degrees is a special angle for which the values of the sine, cosine, and tangent
ratios can be expressed in terms of surds. cos of 45 degrees is exactly equal to
root two over two, a value which you need to know and be able to recall
Therefore, we can determine the
value of 𝑥 exactly without using a calculator. 𝑥 is equal to eight multiplied by
root two over two. This fraction can be simplified
slightly as the eight in the numerator and the two in the denominator can both be
divided by a common factor of two. Therefore, it simplifies to four
multiplied by root two.
The horizontal distance between the
base of the ladder and the wall is four root two meters. Remember you need to know and be
able to recall the values of all three trigonometric ratios for an angle of 45