Classify the triangle, then solve: A=60°,B=15°,c=6A=60°,B=15°,c=6A=60°,B=15°,c=6.

A S A , a = 5.38 , b = 1.61 , C = 105 ° ASA,a=5.38,b=1.61,C=105\degree A S A , a = 5.38 , b = 1.61 , C = 105°

Classify the triangle, then solve: A = 60 ° , B = 15 ° , c = 6 A=60°,B=15°,c=6 A = 60° , B = 15° , c = 6 .

Everyone, welcome back. So in this practice problem, we're gonna classify this triangle given the information of the angles and sides, and then we're gonna solve it. We're just basically gonna figure out all the missing angles and sides that we need. Alright. Let's get started. Now regardless of the type of triangle that you're dealing with, even if you don't know what it is from the beginning, the first thing you're gonna do is you're gonna just gonna try to draw the triangle based on the given information. And I wanna warn you that the way that you might draw your triangles might be different than mine. I'm just gonna walk you through how I sort of think about this, and it really just sort of comes down to looking at the angles because those are the sort of easiest things to visualize in your triangle. So here's what I mean. Right? So I know that angle a is gonna be 60 degrees, and 60 degrees kind of look something like this. I'm gonna draw a line like this and something that looks like that. Right? This looks like a 60 degree angle. Alright? So the next thing I know is that there's another angle in this triangle that's gonna be 15 degrees. 15 degrees is gonna be a really, really small sort of angle here. So I have a couple of options. I can try to draw it off of this little corner over here, but that's gonna make my triangle really small. So what I'm gonna do here is I'm gonna take this, and this is gonna be my 15 degree angle. Right? And that basically sort of connects my triangle now. So here, what I have is 60 degrees, and this is gonna be a. This is gonna be b, which is the 15 degrees over here. And that means that by default, this is angle c. Right? So this is just, again, one plausible way that you could have drawn this triangle. Yours might look a little bit different. Alright? So if this is a, b, and c, that means that this is side length c, and we already know what that is. That's equal to 6. And therefore, what happens is now we've sort of labeled all of the given information that we have. So can we classify this triangle? We actually can because see how we have an angle and an angle. So that's 2 angles, and we have the side that goes in between them. So this is gonna be an angle side angle triangle. So this is an ASA. Alright? So that's the classification. How do we solve it? We actually just solve all these problems really the same way, which is the second step over here, and that we're done with step 1, is we're just gonna use the angle sum formula, the fact that all angles add up 280 to solve for this third angle. Alright? So, basically, because we already have 2 of the angles of the triangle, we can always just figure out that third one just by subtracting both of them from 180. So the step 2 over here is I'm gonna take a, b, c, and I'm gonna add them up, and I'm gonna equal a 180 degrees. And And I'm trying to figure out angle c, so this is just gonna be 180 minus the other 2. 180 degrees minus 60 minus the other one, which is 15. And if you work this out, what you're going to get is that c is equal to a 105 degrees. So that's one of our sort of missing values. So we know that this thing over here is going to be a 100 and 5 degrees, and now we actually have what all of the angles in our triangle are. We know this is 60, this is 15, and this is 105. Alright? How do we solve for the other missing information? Remember, the side opposite of b is going to be b, the side opposite of angle a is going to be a. Alright? So, basically, I'm still have 2 I have 2 variables that I still need, the little a and little b. How do I solve for those? That brings us to the 3rd step, which is basically we're gonna use the law of sines to solve those remaining two sides. And in this case, because there's 2 missing variables, we're just gonna have to use the law of sines twice. Alright? So how do we do this? Well, in step 3, what I'm going to do is I'm just going to start my law of sines formula. Sine of a over a equals sine of big b over b equals sine of big c over c. Again, so we know here that angle A, we actually know what all of the angles are. And what I already know is side actually, I know what side C is equal to. So So, basically, I have an option here. I could either pick these two terms to solve for little b or this one and the first one and solve for little a. It doesn't matter which one you do. That's why it the step 3 just says solve the remaining sides in no particular order. Alright? So what I'm gonna do here is I'm just gonna go ahead and pick these two variables, and I'm gonna solve for, b in this in this equation over here. Okay? So what I'm gonna do here is to bring this down again, I'm just gonna flip this. That's the easiest way to get the b on top. What you're gonna see here is that b is equal to, this is gonna be sine of big b times little c over sine of big c. Alright. That's just what happens when you flip these two equations and then bring the sign over to the other side. So if I just plug in some variables, we're going to see is that big little b is equal to the sign of big b, which big b is equal to the 15 degrees. So 15 times, and then this is gonna be little c, which is the 6 that I know, divided by sine of big c, which is a 105 degrees that I figured out earlier in the problem. So if you work this out, what you're gonna get for little b is you're gonna get something that is 1.61. So 1.61. So that means that this little side over here is 1.61. Notice that that actually kind of makes a lot of sense because, again, what happens is that the angles are usually proportional to the distances on the other side. This is the largest angle, which is 105 degrees, therefore, sort of corresponding to the largest side, which is 6. This one is a really, really tiny angle, 15, and it corresponds to the shortest side, which is 1.61. So that totally makes sense. All right? So that's almost done here. I just have to find out one additional variable, which is now I just have to go ahead and solve for a. Now, again, it doesn't matter which way you do this. You could compare these 2 now that you have basically everything else in the problem or these 2. It really, really doesn't matter. One just tiny little thing that I would recommend is to try and stick with the variables that you already know from the beginning of the problem because when you use these, these are rounded values, and it may throw off your answers the more you round throughout the problem. Alright. So what I'm gonna do here is to calculate a, I'm gonna set up the exact same thing. I'm gonna do sine of a over a, and I'm just going to shortcut this. This is going to equal the sine of c over c, and I'm basically going to set up the equation that solves for little a. Alright? We'll do the exact same thing I did over here. I'm just going to grab this. This is going to be a over sina equals c over sine big C. And I'm just going to move this to the other side and then multiply. So, in other words, a is equal to this is going to be sine of angle a. Angle a, we figured out, was 60 degrees. So this is going to be the sine of 60 times little c, which is the 6 divided by the sine of the 105 that we figured out. Alright. So plugging this in, what you're going to get here is you're going to get a is equal to 5.38. Alright. So that's the other value. Again, come back here to the triangle. And again, we sketch this out, but we can see here that this side over here is 1.61, really tiny. This thing is 6, and this thing is not quite as long. This is going to be 5.39 38 that we just found. That actually totally makes sense because it's almost as long as that other side. So that means that we're done. We figured out all of the remaining sides and we fully solve this triangle. Let me know if you have any questions and thanks for watching.

Classify the triangle, then solve: A = 60 ° , B = 15 ° , c = 6 A=60°,B=15°,c=6 A = 60° , B = 15° , c = 6 .

A S A , a = 5.38 , b = 1.61 , C = 105 ° ASA,a=5.38,b=1.61,C=105\degree A S A , a = 5.38 , b = 1.61 , C = 105°

S A A , a = 6.69 , b = 22.4 , C = 105 ° SAA,a=6.69,b=22.4,C=105\degree S AA , a = 6.69 , b = 22.4 , C = 105°

A S A , a = 6.69 , b = 22.4 , C = 105 ° ASA,a=6.69,b=22.4,C=105\degree A S A , a = 6.69 , b = 22.4 , C = 105°

S A A , a = 5.38 , b = 1.61 , C = 105 ° SAA,a=5.38,b=1.61,C=105\degree S AA , a = 5.38 , b = 1.61 , C = 105°

13. Non-Right Triangles

Law of Sines

Precalculus

13. Non-Right Triangles

Law of Sines

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Multiple Choice

Classify the triangle, then solve: $A=60°,B=15°,c=6$ .

$SAA,a=6.69,b=22.4,C=105\degree$

$ASA,a=6.69,b=22.4,C=105\degree$

$ASA,a=5.38,b=1.61,C=105\degree$

$SAA,a=5.38,b=1.61,C=105\degree$

Verified step by step guidance

Identify the given angles and side: A = 60°, B = 15°, and c = 6. This is a triangle with two angles and one side given, which suggests using the Angle-Side-Angle (ASA) or Side-Angle-Angle (SAA) method.

Calculate the third angle C using the angle sum property of triangles: A + B + C = 180°. Substitute the known values to find C.

Use the Law of Sines to find the unknown sides a and b. The Law of Sines states that $ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $.

Substitute the known values into the Law of Sines to solve for side a: $ a = \frac{c \cdot \sin A}{\sin C} $.

Similarly, solve for side b using the Law of Sines: $ b = \frac{c \cdot \sin B}{\sin C} $.

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