Find the main trigonometric ratios of angle 𝐵, given 𝐴𝐵𝐶 is a right triangle at 𝐶, where 𝐴𝐵 equals 30 centimeters and 𝐵𝐶 equals 18 centimeters.
So we have a right triangle, where it’s a right angle at 𝐶. So we could put 𝐴 and 𝐵 in either corner that remains. 𝐴𝐵 is 30 centimeters and 𝐵𝐶 is 18 centimeters. So when it asks us to find the main trigonometric ratios at 𝐵, we’re finding the sine of 𝐵, the cosine of 𝐵, and the tangent of 𝐵. And sine, cosine, and tangent are abbreviated here. So again, we’re referencing everything to angle 𝐵. However, we’re missing one side of the triangle, side 𝐴𝐶. We can call it 𝑥. And we can use the Pythagorean theorem to find this missing side because it’s a right triangle.
So the square of the longest side, which is 𝐴𝐵, is equal to the square of the two shorter sides, 𝑥 and 18. So we can have 𝑥 squared plus 18 squared equal to 30 squared, and then we can solve for 𝑥. So let’s first go ahead and square 18 and 30. And now, let’s go ahead and subtract 324 from both sides of the equation. So 𝑥 squared is equal to 576. Now to solve for 𝑥, let’s go ahead and square root both sides. And when we take the square root, it would be positive 24 and negative 24. However, this is a length and the length is always positive. So we’re going to use 24 as our length for 𝐴𝐶.
So now working with the trig ratios — sine, cosine, and tangent — there’s a helpful way they can help us remember what they each stand for. The capital letters stand for sine, cosine, and tangent. And then the lower case letters stand for what goes in the ratios, the fractions. So sine is equal to the opposite side divided by the hypotenuse side. The cosine is equal to the adjacent side divided by the hypotenuse side. And then tangent is equal to the opposite side divided by the adjacent side. So if we’re paying attention to angle 𝐵, side 𝐴𝐶 is the side that’s opposite. No matter what angle we’re looking at, the hypotenuse is always the longest side, which is the one across from the 90-degree angle. And then lastly, the adjacent side is the one that’s next to the angle. And again, the hypotenuse is next to it, but the hypotenuse is always in the exact same spot. So the adjacent can be moved depending on which angle we’re looking at, but it’s always the one that’s next to the angle. That’s what adjacent means, touching.
So let’s begin with the sine of 𝐵. Sine is opposite over hypotenuse, so 24 divided by 30. Cosine is equal to adjacent over hypotenuse, so 18 over 30. And then tangent is equal to opposite over adjacent, so 24 divided by 18. However, each of these ratios can be reduced. Every single number can be reduced by six, so we can divide every number by six. So twenty-four thirtieths turns into four-fifths, eighteen thirtieths turns into three-fifths, and twenty-four eighteenths turns into four-thirds.
Therefore, the sine of 𝐵 is equal to four-fifths, the cosine of 𝐵 is equal to three-fifths, and the tangent of 𝐵 is equal to four-thirds.