𝐴𝐵𝐶 is a triangle where 𝑎 equals 25 centimeters, 𝑏 equals 20 centimeters, and 𝑐 equals 29 centimeters. Find the measure of angle 𝐴 giving the answer to the nearest second.
We know that 𝐴𝐵𝐶 is a triangle. And we’re given the three side lengths of this triangle. All three of the side lengths have a different measure, making this triangle scalene. If we sketch a triangle, the longest side is side 𝑐, which has a measure of 29 centimeters. The next longest side is 𝑎, which has a measure of 25 centimeters. And the remaining side length is 𝑏 at 20 centimeters. We wanna label the vertices with the capital letters such that 𝐴 is opposite the side length lowercase 𝑎, 𝐵 is opposite the side length lowercase 𝑏, and the same for 𝐶. The angle we’re interested in is angle 𝐴. This is the missing angle we want to solve for.
When we know three side lengths in a triangle, how can we go from those three side lengths to calculating a missing angle? We can do that with the law of cosines. The law of cosines says that 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared minus two 𝑎𝑏 times cos of 𝐶. The way that the formula is usually written, it looks like you’re solving for a third side length. However, we know the values of 𝑎, 𝑏, and 𝑐. And that means the only variable in our argument would be the angle 𝐶.
This is where things might be a little bit confusing. And so it can be a good idea to rearrange the equation so the argument we’re solving for is by itself. We want to rewrite the equation so that cos of 𝐶 is by itself. To do that, we subtract 𝑎 squared and 𝑏 squared from both sides of our equation so that we have 𝑐 squared minus 𝑎 squared minus 𝑏 squared equals negative two 𝑎𝑏 times cos of 𝑐. From there, we divide both sides of the equation by negative two 𝑎𝑏. When we divide and simplify, we then have 𝑎 squared plus 𝑏 squared minus 𝑐 squared over two 𝑎𝑏 equals the cos of 𝑐.
But we still need to think critically here. Remember, we’re looking for the measure of angle 𝐴. And this formula is for the measure of angle 𝐶. 𝐶 is the missing angle. And 𝑎 and 𝑏 are the two side lengths adjacent to the angle. 𝑐 is the opposite side length. And that means we need to rearrange the variables we’re using. Since we’re looking for the cos of 𝐴, we need to add 𝑏 squared plus 𝑐 squared. These are the two side lengths adjacent to our missing angle.
And then we need to subtract 𝑎 squared since that is the side length opposite our missing angle and then divide by two 𝑏𝑐, divide that by two times the two adjacent side lengths. We now have the formula in the correct form that we need to find the value of our missing angle 𝐴. Let’s clear some space and plug in what we know.
Using the formula we’ve just found, we can say that the cos of 𝐴 is equal to 20 squared plus 29 squared minus 25 squared all over two times 20 times 29. This gives us that the cos of 𝐴 equals 616 over 1160. Both the numerator and the denominator are divisible by eight. And so we can simplify this to say the cos of 𝐴 equals 77 over 145. At this point, we’ll have to take the inverse cosine to find out the exact value of 𝐴. If we take the inverse cos of the left-hand side of the equation, we need to take the inverse cos of the right-hand side of the equation.
The inverse cos of cos of 𝐴 just equals 𝐴. And the inverse cos of 77 over 145 is 57.924622 continuing. This is a measure of degrees. To give the answer to the nearest second, we take the whole number portion of our degrees. So we have 57 degrees. And then we take our remaining decimal value 0.924622 continuing, and we multiply it by 60. This will give us the number of minutes in this partial degree. When we do that, we get 55.477359 continuing minutes.
We take the whole number value as our minute. 57 degrees, 55 minutes. From there, we take the remaining partial minute and multiply it by 60 to find the number of seconds in that partial minute, which gives us 28.64158 continuing seconds. We’re measuring to the nearest second. 28.64158 continuing seconds will round up to 29. So we can say that the measure of angle 𝐴 is equal to 57 degrees, 55 minutes, and 29 seconds.