Video Transcript
๐ด๐ต๐ถ is a triangle where eight sin ๐ด equals 11 sin ๐ต which equals 16 sin ๐ถ. Find the ratio ๐ to ๐ to ๐.
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, weโll compare the equation weโve been given in the problem to the given formula. Weโll 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. Weโll need to manipulate our equation to make it look like this. To do so, weโll need to divide each of its component parts by some number. Weโll divide here by the lowest common multiple of eight, 11, and 16.
That will mean weโll have sin ๐ด over some number equals sin ๐ต over some number which equals sin ๐ถ 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. Thatโs two to the power of four multiplied by 11, which is 176. So weโll need to divide each part of our equation by 176.
Eight sin ๐ด divided by 176 is sin ๐ด over 22, 11 sin ๐ต over 176 simplifies to sin ๐ต over 16, and 16 sin ๐ถ over 176 becomes sin ๐ถ over 11. We can see now that this equation looks just like the general form for the law of sines.
The ratio of ๐ to ๐ to ๐ then is 22 to 16 to 11. These three numbers are coprime, which means the only factor they share is one. And this ratio canโt be simplified any further.
The ratio of ๐ to ๐ to ๐ is 22 to 16 to 11.
