Apply the law of sines to the following: a = √5, c = 2√5, A = 30°. What is the value of ​sin​ ​C​​? What is the measure​ of ​​​​C​​​? Based on it​​s angle measures, what kind of triangle is triangl​e ​​ABC​​​?

Video transcript

Welcome back. Everyone. In this problem, we want to determine the measure of angle C in the triangle ABC and identify the type of triangle for our triangle. Its measurements are that the side A is six multiplied by the square root of seven units long. The side C is 12 multiplied by the square root of seven units long. And angle A is 30 degrees. For our answer choices A says C is 60 degrees and it's a right scaling triangle B says C is 60 degrees and it's a right ISOS triangle C says it angle C is 90 degrees and it's a right scaling triangle and D says angle C is 90 degrees and it's a right iso triangle. No. In this problem, in order to figure out the measure of angle C, let's first try to do a sketch of the triangle just to get an idea of where the sides and angles would be. OK. So let's say that this is our triangle. OK. Let's say that this is our triangle here. And in our triangle, OK, we know that the angle A, let's put the angle A here is 30 degrees. OK. Let's say that these sides are angle B and sorry, those corners are angle B and C. And in the naming convention for a triangle, the side with the side opposite to the angle is the same letter but common. So if this is angle C, then side C would be opposite to an C which is 12 multiplied by the square of seven and the side A would be opposite to anger A which is six multiplied by the square root of seven. And in this problem, we're trying to solve for angle C now to solve for Anglesey, let's think a little bit about what we know what information do we have here. Well, we have the value of anger A and its opposite side A. OK? And we have the value of side C but we don't know its opposite angle C. So we know a pair of angles and sides. What could we use to help us figure out our unknown angle. C given that we know its opposite side. Well, we could use the law of science. OK? And recall that by the law of science, by the law of science, OK. Then we know we know so far that the law of science tells us the sine of an angle to its opposite side. That is the ratio of the sine of an angle to its opposite side in a triangle is the same throat. So the sine of A to A is the same thing. As the sine of B to B, which is the same thing as the sine of C to C. Now, in our problem here, we want to figure out an CC is what we want to know. We already know the value of sc and we know the value of anger A and the value of side A. So we can use the both of these to help us to eventually solve for the sign of C. Because by the law of science, it tells us then that the sine of C OK to its opposite side, 12 route seven is going to be equal to the sine of A which is 30 degrees to its opposite side which is six root seven. Now we can solve for the value of C. Now here, if we multiply both sides by 12 root seven, we'll get the sine of C being equal to 12 root seven multiplied by the sine of 30 degrees divided by six, multiplied by the square root of seven or six root seven. Now, here we have some common terms. Route seven cancels route seven and six goes into 12, 2 times. Therefore, the sine of C is equal to a half or sorry is equal to two multiplied by the sine of 30. Now, here's the interesting thing, the sine of 30 equals a half. So therefore, the sine of C equals two multiplied by a half which equals one. So the sine of C equals one. So to figure out the value of C, we're basically asking ourselves the sine of which angle is equal to one. How are you gonna figure that out? Well, we could do two things, you know, first, we could say then that C is going to be equal to the inverse sine of one to figure out what angle that is. And when we do that, we find that C is equal to 90 degrees. And that makes sense because we already know that the sine of 90 degrees equals one. Therefore, our value C our, our angle sea is 90 degrees. So instead it should be looking something like this. OK. That's 90 degrees. Now, the fact that it's 90 degrees tells us that this is a right triangle but no, which one is it? Is it a right scaling triangle or a right isoli triangle? Well, here notice that the angle A is 30 degrees and since it's a right triangle, then angle B must be equal to 60 degrees because 60 plus 30 plus 90 would equal 180 degrees. So since B is 60 degrees A is 30 degrees, both are less than 90 degrees. That means our triangle ABC is a right scaling triangle. So angle C is 90 degrees. This is a right scaling triangle. The correct answer is answer choice. C thanks a lot for watching everyone. I hope this video helped.