Find the length of 𝐴𝐵.
We will answer this question using the right-angle trig ratios and our knowledge of special trig angles. We begin by recalling the acronym SOHCAHTOA, which helps us find the sine, cosine, and tangent ratios in right-angled triangles.
The longest side of a right-angled triangle is known as the hypotenuse. In this question, it is length 𝐴𝐶. Length 𝐵𝐶 is opposite the 60-degree angle. Length 𝐴𝐵 is next to the 60-degree angle and the right angle and is known as the adjacent. This is the length we need to calculate, which we will label 𝑥.
As we are dealing with the adjacent and hypotenuse, we will use the cosine ratio. This states that the cosine or cos of 𝜃 is equal to the adjacent over the hypotenuse. Substituting in our values, we have cos of 60 degrees is equal to 𝑥 over 7.8. We now need to recall some of our special trig angles. Sin of 60 degrees is root three over two. Cos of 60 degrees is one-half. And tan of 60 degrees is equal to root three.
Replacing cos of 60 degrees with one-half gives us one-half is equal to 𝑥 over 7.8. Multiplying both sides of this equation by 7.8 gives us 7.8 multiplied by one-half equals 𝑥. One-half of 7.8 is 3.9.
We can, therefore, conclude that the length of 𝐴𝐵 is 3.9 centimeters. In any right-angled triangle with a second angle of 60 degrees, the side adjacent to this will be half the length of the hypotenuse.