Use the law of sines to prove that each statement is true for ...

Question

Use the law of sines to prove that each statement is true for any triangle ABC, with corresponding sides a, b, and c.

(a - b)/(a + b) = (sin A - sin B)/(sin A + sin B)

Accepted Answer

Welcome back. Everyone. In this problem, we want to determine whether the statement A plus B divided by A minus B equals sine A plus sine B divided by sin, A minus sine B is true or false. For our answer choices A says it's true. B says it's false. Now, if you notice here, OK, we have our expression, our values for A B and we have S A and the sine B in our uh formula here in our statement. So what we need to ask ourselves is do we know anything that relates the sine of A and the sine of B to the side A and B in a triangle? Well, we can recall the law of science. OK? Let me put that in red here. And if we recall the love of science, the love of science basically tells us that for a triangle with the sides A B and C and angles A B and C. In other words, a triangle that looks like this. OK? Then the s of an anger, the ratio of the sine of an anger to its opposite side in the triangle is the same throughout the triangle. In other words, the law of science tells us that the ratio of side A to sine A is the same thing as side B to side B, which is the same thing as side C to the S of C Angus C. And all of these are the same ratio. So in other words, these equals home value K. Now, if we want to prove if the statement is true or false, we will need to express a in terms of sine A and express B in terms of sine B. And we can use this law to help us. Because what that means is that if all of these equal K, then A divided by sine A equal care, which means that A is going to be equal to K multiplied by sine A. So we've expressed A in terms of sine A. Likewise, we can also say then that B to sine B is going to be equal to K, which means that B is going to be equal to K multiplied by the sine of B. Now that we have A and B in terms of sine A and sine B respectively, we can substitute that into our expression to see if we can eventually break it down to sin A plus sine B divided by sine A minus sine B. So let's try to do that. So now our first expression A plus B or, or, or left hand side A plus B divided by A minus B can be written as K sine A, OK, plus K sine B divided by K sine A minus K sine B. Now notice all of these terms have K in common. So we can factor K from the numerator and the denominator from the numerator. That means we're going to be left with K multiplied by the sine of A plus the sine of B. We've, you know, factored out K and in the denominator factoring, OK. We're going to have K multiplied by the sine of A minus the sine of B. Now, what do we notice here? K is in the numerator and the denominator so we can cancel K and when we cancel K, now we get our expression to be the sine of A plus the sine of B divided by the sine of A minus the sine of B. Therefore, A B divided by A minus B is indeed equal to the sine of A sine B divided by sine A minus sine B. Our statement is true. The correct answer is choice. A thanks a lot for watching everyone. I hope this video helped.

Use the law of sines to prove that each statement is true for any triangle ABC, with corresponding sides a, b, and c.

(a - b)/(a + b) = (sin A - sin B)/(sin A + sin B)

Key Concepts

Law of Sines

Trigonometric Identities

Algebraic Manipulation

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