𝐴𝐵𝐶 is a triangle where eight sin 𝐴 equals 11 sin 𝐵 which equals 16 sin 𝐶. Find the ratio 𝑎 to 𝑏 to 𝑐.
Remember the law of sines tells us that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and all angles. Rather than sketch this triangle out, we’ll compare the equation we’ve been given in the problem to the given formula. We’ll choose this first formula.
Since in our equation, the sine of each of the angles is not on the denominator of a fraction. Notice how in the general form, the lengths of the sides are represented by the denominator of each fraction. We’ll need to manipulate our equation to make it look like this. To do so, we’ll need to divide each of its component parts by some number. We’ll divide here by the lowest common multiple of eight, 11, and 16.
That will mean we’ll have sin 𝐴 over some number equals sin 𝐵 over some number which equals sin 𝐶 over some number. And it will look like the general form of the equation. The highest common factor of eight, 11, and 16 can be found by considering each number as the product of its prime factors. Eight is two cubed, 11 is simply 11, and 16 is two to the power of four.
The lowest common multiple of these numbers can be found by calculating the product of the largest multiple of each prime that appears on at least one list. That’s two to the power of four multiplied by 11, which is 176. So we’ll need to divide each part of our equation by 176.
Eight sin 𝐴 divided by 176 is sin 𝐴 over 22, 11 sin 𝐵 over 176 simplifies to sin 𝐵 over 16, and 16 sin 𝐶 over 176 becomes sin 𝐶 over 11. We can see now that this equation looks just like the general form for the law of sines.
The ratio of 𝑎 to 𝑏 to 𝑐 then is 22 to 16 to 11. These three numbers are coprime, which means the only factor they share is one. And this ratio can’t be simplified any further.
The ratio of 𝑎 to 𝑏 to 𝑐 is 22 to 16 to 11.