The law of sines The law of sines says that if a, b, and c are...

Question

The law of sines The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a triangle, then

(sin A) / a = (sin B) / b = (sin C) / c

Use the accompanying figures and the identity sin (π − θ) = sin θ, if required, to derive the law.

tri

Accepted Answer

Welcome back, everyone. In this problem, using the given figure, a cosine function, and the Pythagorean theorem derive the law of cosine equation for A. Here we have our figure, and for our answer choices, A says A2 equals B + C2 minus 2BC + B. B says A2 equals B + C2 minus 2BC A. C says A2 equals B minus C2BC + B, and the D says A equals B2 minus C2 minus 2BC A. Now, how can we use this figure and the associated rules to help us derive the law of the cosine for the law of the cosine equation for A. Well, the first thing we can do here, notice that we have our figure. It's a triangle, but really it's split into two right triangles, OK? So if we label this point where our vertical line meets AC as point D, OK. Let me just put that in red so it's noticeable. OK. And if we label here this length from A to D as X, then we can call from D to C or segment CD as B minus X. OK? I'm trying to find a creative way to put that here. Uh, let me move, let me just move D for a minute. OK, and oh, you know what, let's keep D, but let's just use a little arrow to show that this line segment is B minus X, OK? So that's called that B, the difference between B and X. Now if we think about it here from these two right triangles we could apply the Pythagorean theorem to relate their height H or the height H. So how can we do that? We'll recall that the Pythagorean theorem, OK, Pythagorean theorem tells us that the square of two sides, OK, let's say for a triangle X2 + Y2. The, the square of the two shorter sides. The sum of the squares of the two shorter sides equals the square of the longest side, the hypotens, OK? So, in that case, if we apply that to our triangles, 4 triangle A, B, D, by the Pythagorean theorem, OK, then 8 squared. Plus X2 will be equal to C2, which means that 82 will be equal to C2 minus X2. Likewise, for triangle BCD, OK. Then we will have H2, OK, plus B minus X2 equal to A2. And in that case, that means H H2 equals A2, OK. Minus B minus X2. Now, in that case, since we have both of these or we solved for H2d from ABD and BCD, then we, we can equate both of these equations. In other words, this tells us then that C2 minus X2 is gonna be equal to A2 minus B minus X2. Now if we do some expanding here, OK, then. We should find that we have here A2 minus, OK, let's expand. This is gonna be B2 minus 2BX plus X2, OK? And if we subtract all of that, then we're gonna get A2 minus B2 + 2BX, OK? minus X2. All of that will be got the C2 minus X2. Notice here that we can cancel minus X2 from both sides, OK, to get C2. Equal to a minus B2 + 2B X. And now, since we're solving for A2, that means A2 equals C2 + B2 minus 2BX. Now we're almost there in terms of solving for A2. We have C2 and B2, but here we have X and X was a value that we put into our figure. So how can we get the soul to express the law of cosines purely in terms of A B and C2 or in terms of its angles A, B, and C? Well, let's go back to our figure. Notice that from triangle ABD, OK, we could apply the cosine rule. Because from this part of, from our triangle here, let's highlight that in red. So, from triangle ABD, OK. Then notice here that by the cosine rule. Let's make a note here. So by the cosign rule. by the cosine ratio, what we know about the cosine ratio. We could say that a cosine of A is equal to X divided by C. Because remember, the cosine is the ratio of the adjacent side to the hypotenuse, OK? So in this case for our angle A. The cosine of its adjacent side X, the cosine of A will be the adjacent side X divided by the hypotenuse C. Now why did we do that? Well, no, that means we can express X as C A. C multiplied by A. And if that's the case, if we put that into our formula, then A2 equals B2 plus C2 minus 2B multiplied by X which is C plus A. Thus, this is the law of cosine's equation for a. If we look back at our answer choices, that means answer choice B is the correct one. Thanks a lot for watching everyone. I hope this video helped.

The law of sines The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a triangle, then

(sin A) / a = (sin B) / b = (sin C) / c

Use the accompanying figures and the identity sin (π − θ) = sin θ, if required, to derive the law.

Key Concepts

Law of Sines

Trigonometric Identities

Triangle Properties

Watch next