Find the length of 𝐵𝐶 in the figure to two decimal places.
In order to answer this question, we need to recall the rules of right-angle trigonometry together with the special trig angles. In this question, we need to calculate the length of 𝐵𝐶, which we will call 𝑥. Next, we remember the acronym SOHCAHTOA. This deals with the sine, cosine, and tangent ratios in right-angled triangles. The longest side of a right-angled triangle is opposite the right angle. This is called the hypotenuse. Length 𝐴𝐵 is the opposite side as it is opposite the 30-degree angle we are working with. Length 𝐵𝐶 is the adjacent as it is next to the 30-degree angle and the right angle.
As we are dealing with the opposite and adjacent in this question, we will use the tan ratio. This states that tan of 𝜃 is equal to the opposite over the adjacent. Substituting in our values, we have tan of 30 degrees is equal to six over 𝑥. We now need to recall our special trig angles. Sin of 30 is one-half, cos of 30 is root three over two, and tan of 30 is root three over three. This is also sometimes written as one over root three.
In this question, we will use the second form such that one over root three is equal to six over 𝑥. Cross-multiplying here means that we multiply both sides by 𝑥 and root three. 𝑥 is therefore equal to six root three. As we need to work out our answer to two decimal places, we need to type this into the calculator. 𝑥 is equal to 10.3923 and so on. The deciding number is the two in the thousandths column. If this number is less than five, we round down. We can therefore say that 𝐵𝐶 is equal to 10.39 units.