In the given figure, find the length of 𝐴𝐶 to two decimal places.
Our first step here is to let the length 𝐴𝐶 equal 𝑥. We can next label our triangle and use right angle trigonometry to work out the missing length. Right angle trigonometry involves the sine, cosine, and tangent ratios often known as SOHCAHTOA.
The length 𝐴𝐶 is the hypotenuse as it is the longest side and is opposite the right angle. The length 𝐴𝐵 is the opposite as it is opposite the 40-degree angle we will be working with. Finally, the length 𝐵𝐶 is the adjacent as it is adjacent or next to the 40-degree angle and the right angle. We know the length of the adjacent. And we’re trying to calculate the hypotenuse. This means that we will use the cosine ratio.
The cosine ratio states that cos 𝜃 is equal to the adjacent divided by the hypotenuse. Substituting in the values from our diagram gives us cos 40 is equal to five divided by 𝑥. Multiplying both sides of this equation by 𝑥 and then dividing by cos 40 means that the cos 40 and 𝑥 will in effect swap places. This means that 𝑥 is equal to five divided by cos 40.
At this stage, it’s important to remember that the hypotenuse was the longest side. Therefore, our answer needs to be greater than five. Typing five divided by cos 40 into our calculator gives us an answer of 6.5270 and so on.
We were asked to give our answer to two decimal places. This means that our answer must have two numbers after the decimal point. The deciding number is the seven. And as this is greater than five, we will round up. 𝑥 is therefore equal to 6.53 to two decimal places.
As 𝑥 was the length 𝐴𝐶 on the triangle, we can say that 𝐴𝐶 is equal to 6.53. There’re no units in this question. But this could be any units of length, for example, centimeters or meters.