𝐴𝐵𝐶 is a triangle where the measure of angle 𝐵 is 110 degrees, 𝑏 equals 16 centimetres, and 𝑐 equals 12 centimetres. How many possible solutions are there for the other lengths and angles?
Remember, here lowercase 𝑏 and lowercase 𝑐 refer to sides of the triangle. Let’s first draw a quick sketch so we can visualise the triangle more easily. So the triangle looks something like this.
Now with this convention of labelling sides and angles with uppercase letters for angles and lowercase letters for sides, remember side 𝑐 is opposite angle 𝐶, side 𝑎 is opposite angle 𝐴, and so on.
So I have an angle of 110 degrees. Opposite that, we have the side of 16 centimetres. And then we also have side 𝑐, which is 12 centimetres. The question asks us how many possible solutions are there for the other lengths and angles in this triangle.
Now we’ll have a little think about what this means a little bit later on. But Let’s begin by thinking about how we could calculate any of the angles and lengths in the triangle if it is possible to do so.
Within the information we’ve been given, we have an opposite pair. So an angle and the side opposite are both known. This suggests that we can use the law of sines within this question. What would we be able to calculate with the law of sines? Well given that we know the side of length 12 centimetres, we’ll be able to calculate the opposite angle, angle 𝐶.
Remember, the law of sines tells us that within a particular triangle, which does not have to be right-angled, the ratio between the sine of each angle and then the length of the opposite side is constant. You may be used to seeing this law of sines written the other way up, with the sides in the numerator.
But as it’s an angle that I’m looking to calculate here, I’m going to use this reciprocal version where the angles are in the numerator. In practice, we only use two parts of this ratio together. So here we’re going to be using the part involving 𝑏s and 𝑐s. Lets substitute in the known values for side 𝑏, side 𝑐, and angle 𝐵.
We have that sine of angle 𝐶 divided by 12 is equal to sine of 110 degrees divided by 16. Now let’s see if we can solve this equation for angle 𝐶. We’ll begin by multiplying both sides of the equation by 12. This tells us that sine of angle 𝐶 is equal to 12 sine of 110 degrees divided by 16.
Now as a decimal, this means that sin 𝐶 is equal to 0.70476. And I’ve evaluated that using my calculator and kept the value on the calculator display. In order to find angle 𝐶, I now need to use the inverse sine function. So using my calculator to evaluate this tells me that 𝐶 is sine inverse of this decimal 0.70476. And to one decimal place, this is 44.8 degrees.
So let’s think about what we’ve done. We’ve applied the law of sines to find a value for angle 𝐶, 44.8 degrees. We could also then calculate angle 𝐴 using the fact that the angle sum in a triangle is a 180 degrees. So we have that angle 𝐴 is equal to 25.2 degrees.
We could then apply the law of sines again using a different pair, this time the pair of 𝑏s and 𝑎s, in order to calculate the third side of the triangle, side 𝑎. We’re not actually going to do that because the question doesn’t actually ask us to find the other lengths and angles. It just asks for the possibilities.
So we know there is at least one possible solution for the other lengths and angles. The question is are there any more. The possibility of there being more arise at this stage here, where we have that sine of 𝑐 is equal to 0.704. And the reason for this is that there are two angles less than 180 degrees that have the same sine. It’s a general rule that for values of 𝑐 less than 180, sine of an angle 𝐶 is the same as the sine of 180 minus 𝐶.
Therefore, it’s possible that another value of 𝐶 is 180 degrees minus our calculated value of 44.8, which would be 135.2 degrees. However, remember we already know that angle 𝐵 is 110 degrees. If angle 𝐶 were 135.2 degrees, this would mean that the sum of these two angles alone would be 245.2 degrees, which would exceed the angle sum in a triangle of 180 degrees.
Well this means then is because of the information we already know about angle 𝐵, it isn’t possible for angle 𝐶 or indeed any of the other unknowns to take a different value from the ones we’ve just calculated. So in answer to the question, there is just one possible solution for the other lengths and other angles in this triangle.
Now you may be wondering are one or two possible solutions the only potential answers to this question. And in fact, they’re not because there is a third possibility. Remember we found that sine of 𝐶 was equal to 0.70476. Sometimes in questions like this, the values you’ve been given may lead you to something like sine of 𝐶 is equal to 1.24.
And if that’s the case, then there are no possible solutions for the other lengths and angles. Why? Well it’s because sine of an angle is always between negative one and one. And in fact, for an angle that’s between zero degrees and 180 degrees, sine of that angle will always be between zero and one.
Therefore, if you have a question similar to this and the working leads you to sine of 𝐶 is equal to something greater than one, then you know that there are no possible solutions because that equation can’t be solved to find a value for 𝐶.
Remember, our answer for this problem with the set of angles and sides that we’ve been given is that there is one possible solution for the remaining lengths and angles in this triangle.