𝐴𝐵𝐶 is a triangle where 𝑎 equals 13 centimetres, 𝑏 equals 10 centimetres, and cos of 𝐶 equals 0.2. Find the value of 𝑐, giving your answer to three decimal places.
First of all, there’s an important distinction to be made here between the uses of the letter 𝑐. Remember lower case letters represent sides and uppercase letters represent angles. First, let’s sketch this triangle to help visualise the problem a little more clearly.
The triangle looks something like this. So we’ve been given the lengths of two the sides and some information relating to the size of the included angle. We’ve then been asked to calculate the length of the third side. All of this information suggest that we need to use the law of cosines for this problem.
Now the law of cosines, remember, is this: 𝑎 squared is equal to 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 cosine of 𝐴, where the lower case letters represent sides and the uppercase 𝐴 represents an angle. However, this former isn’t particularly useful here as it isn’t cosine of angle 𝐴 that we’ve been given, it’s cosine of angle 𝐶.
So I’m going to rewrite this but I’m going to swap 𝐴s and 𝐶s around. So everywhere that there’s a lowercase 𝑎, I’m gonna replace it with a lowercase 𝑐. Lowercase 𝑐s will be replaced with lowercase 𝑎s. And the uppercase 𝐴 will be replaced with an uppercase 𝐶.
This gives me an alternative specification of the law of cosines, but where it’s 𝑐 squared that we’re calculating. And it tells me that 𝑐 squared is equal to 𝑏 squared plus 𝑎 squared minus two 𝑏𝑎 cosine of 𝐶.
Remember sides 𝑎, 𝑏, and 𝑐 are the sides opposite the angles with the same letter. So I’ve marked them in orange on the diagram. Now we’re ready to substitute the lengths of sides 𝑎 and 𝑏 and the value of cos 𝐶 in this question.
So we have that 𝑐 squared is equal to 10 squared plus 13 squared minus two multiplied by 10 multiplied by 13 multiplied by cos of 𝐶, which remember is 0.2. Evaluating each of these terms gives 𝑐 squared is equal to 100 plus 169 minus 52.
This simplifies to 𝑐 squared is equal to 217. Next, in order to find the value of 𝑐, we need to take the square root of both sides of this equation. So 𝑐 is equal to the square root of 217.
And as a decimal, this is 14.730919. The question has asked for this value to three decimal places. So we need to round it. So we have that the value of 𝑐 to three decimal places is 14.731 centimetres.