Question Video: Using the Cosine Ratio to Find Lengths in Right-Angled Triangles Mathematics

Video Transcript

Find the length 𝐴𝐵, giving your answer to two decimal places.

The length 𝐴𝐶 in the diagram is 39 centimetres. And the angle 𝐵𝐴𝐶 is equal to 37 degrees. As the triangle is right-angled, we can solve this problem using the trigonometrical ratios. Sin 𝜃 is equal to the opposite divided by the hypotenuse. Cos 𝜃 is equal to the adjacent divided by the hypotenuse. And tan 𝜃 is equal to the opposite divided by the adjacent.

In this example, we’re looking to calculate the length 𝐴𝐵, labelled 𝑥 on the diagram. The length 𝐴𝐶 is the hypotenuse of the triangle as it is the longest side and it’s opposite the right angle. 𝐵𝐶 is the opposite as it is opposite the 37-degree angle. And 𝐴𝐵 is the adjacent as it is next to or adjacent to the 37- and 90-degree angles.

As we are going to use the adjacent and the hypotenuse, we’ll use the cosine ratio. Cos 𝜃 equals the adjacent divided by the hypotenuse. Substituting in our values from the diagram gives us cos 37 is equal to 𝑥 divided by 39. Multiplying both sides of this equation by 39 gives us 39 multiplied by cos 37 is equal to 𝑥. This gives a value of 𝑥 of 31.15. This means that the length of 𝐴𝐵 is 31.15 centimetres to two decimal places.