Video: Using Right-Angled Triangle Trigonometry to Find Lengths in Word Problems

A ladder 8 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°.

03:49

Video Transcript

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 metres.

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 yourself.

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 degrees.

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