Question Video: Using Trigonometric Ratios to Find All the Angles in Right-Angled Triangles Mathematics

Video Transcript

Find the values of 𝛼 and 𝛽, giving your answer to the nearest second.

In this right triangle, we’re missing these two angle measures. We’re given the lengths of the two smallest sides in the triangle. We don’t know the length of the side opposite the right angle, which is the hypotenuse. Because we’re not given the length of the hypotenuse, I immediately think of the tangent ratio. To use tangent ratio, we don’t need the length of the hypotenuse. We know that tangent of an angle is equal to the opposite side length over the adjacent side length.

Here the tangent of angle 𝛼 is equal to 33 over 46, opposite over adjacent. In order to isolate this angle measure, to get it by itself, we’ll need to take the tangent inverse of tangent 𝛼. The tangent inverse of tangent 𝛼 equals the angle measure, but we need to keep this equation balanced. And if we take the tangent inverse on the left side, we also need to take the tangent inverse on the right side.

Angle 𝛼 is equal to the tangent inverse of 33 over 46. We’ll use a calculator or a computer to solve the tangent inverse of 33 over 46. Your calculator is going to return one of two things when you plug this in: it could tell you that the tangent inverse of 33 over 46 is equal to 0.6223028712; the second option it might give you 35.6553280979. The calculator on the left that gave us about 62 hundredth is operating in radians. The calculator on the right is set to degrees.

Our instructions are to round to the nearest second, and seconds are a subset of degrees. We’re going to have to use a calculator that’s set to degrees. What should you do if your operating with a calculator that set to radians? Look in your general settings, and change the angle unit to degrees. Once you do that, you should get the inverse tangent to be about 35.66 degrees. Now that our calculators are all set, we can move forward.

Now that we found out that angle 𝛼 measures 35.6553 continuing degrees, how do we take this value and round it to the nearest second? We need to think about what we know about degrees. We have degrees. The next smallest unit of measure is minutes. One degree equals 60 minutes. The minute symbol is a little tick mark. And after minutes comes the seconds. One minute equals 60 seconds. And we mark our seconds with two tick marks.

Our task is then to take this value, 35.6553, and convert it into degrees, minutes, and seconds. The degree part is fairly straightforward. The whole number 35 is the number of degrees in this angle. We have to use this decimal value, 0.6553, and find out how many minutes 0.6553 of a degree is equal to. To take this fraction of a degree and convert it into minutes, we’ll need to multiply the fractional value by 60, because there are 60 minutes in every degree.

After multiplying this fractional value by 60, our calculator returns to us 39 and 319 thousandths: 39.318. We’ll follow the same process here that we did with the degrees. The number of minutes will be the whole number, 39. And then we’ll calculate how many seconds is 318 thousandths of a minute. So we take this decimal value and multiply it by 60, because there are 60 seconds in every minute.

0.318 times 60 equals 19.08 of a second. So 318 thousandths of a minute is equal to 19 seconds and eight hundredths of a second. We’re rounding to the nearest second. 19 and eight hundredths rounded to the nearest second is equal to 19. And that tells us that the measure of angle 𝛼 is equal to 35 degrees 39 minutes and 19 seconds.

Now we can repeat this process over again in the exact same way to find the measure of angle 𝛽. However, because we know that inside a triangle all the angles add up to 180 degrees, we can use this information and subtraction to solve for angle 𝛽. This triangle has a right angle, which measures 90 degrees. If you add angle 𝛼 and angle 𝛽 together, they must be equal to 90 degrees. If we take 90 degrees and subtract the measure of angle 𝛼, it will give us the measure of angle 𝛽.

We need to subtract 35 degrees 39 minutes and 19 seconds from 90 degrees. But what do we do here? We need to borrow from this 90 degrees to give us minutes and seconds. If I borrow one degree from 90 degrees, I’ll have 89 degrees. One degree is equal to 60 minutes. If I take one minute away from 60 minutes, I’ll have 59 minutes. One minute equals 60 seconds. 89 degrees 59 minutes and 60 seconds is the same thing as 90 degrees written in a different format.

We can use this format to subtract. Starting all the way on the right, we subtract 19 from 60, which equals 41 seconds. 59 minus 39 equals 20 minutes. 89 minus 35 equals 54 degrees. Angle 𝛽 is equal to 54 degrees 20 minutes and 41 seconds.