Use the Law of Sines to find the length of side a a a to two decimal places .

everyone. Let's take a look at this practice problem. Hopefully, you had a chance to work it out on your own. We have a non right triangle here, and we have to solve for this missing side a. Remember, we can't use SOH CAH TOA, and we can't use the pythagorean theorem. Those things just don't work. So we're gonna have to use this new equation, the law of sines, to calculate this missing side over here. Let's just get started. Now whenever you deal with these types of problems, the just first thing you should do is always actually just write the law of sines out. So it's gonna be the sine of angle a over side a equals the sine of angle b over side b. That's equal to the sine of angle c over side c. Right? So it's always sines, and then all the angles are over their respective sides. Alright? So remember the whole thing here is we're gonna try to get 2 out of these 3 terms or ratios in which we know 3 out of 4 variables. Alright? So So if I take a look through this problem, I've got angle a, which is 45 degrees. I already know what that is. Angle b, which is 75 degrees. And in terms of the sides, I don't know what a is. In fact, that's actually what I'm trying to find here. So I'm gonna have to use one of these other sort of terms. And I know what side b is, but I know nothing about angle c or side c. So remember, we're trying to pick 2 of the 3 ratios. Because I know nothing about c, it's basically like it just doesn't even exist. Alright? So we're just gonna use these two ratios over here. And if you take a look, we actually do know 3 out of 4 variables. We could always solve for that missing one. Alright? So, there's a couple of ways that you could sort of rearrange for this. This is kind of on the denominator, so you could cross multiply, but I actually find that the easiest way to do this is basically just flip the fractions. So if you flip the fraction sign of angle a over a, it becomes a over sine of angle a. Alright? And this is gonna equal we're gonna do whatever you do to one side, you have to do the other. So this is gonna be b over sine of angle b. Alright? So you could totally flip these things. Totally fine. Alright? So, So, basically, what I'm gonna do now is now I'm gonna take this sine a, and I'm gonna move this over here and multiply it out. And I'm just gonna start plugging in some variables because I'm pretty much done here. I know everything else. So a is just going to equal the sine of angle a, which is going to be sine of 45 degrees times and then b over sine b. So little b is gonna be 6 divided by the sign of angle b, which is gonna be 75. Alright. So if you plug this into your calculator, you may wanna sort of stick this in parentheses over here. Just be careful. But what you should see is that you should get something like 4.39. Alright? So 4.39 is gonna be your side length a. Notice how this actually kind of makes sense here because if you look at this angle, this is or sorry. This side this side is 6, and this side is a little bit shorter than that. That's that's quite 6, and this is ends up being 4.39. Also, in general, what happens is there ends up being a portion between the angles. The higher angles usually have the higher sides, so 75 gives us 6. So 45 gives us something that's a little bit less than that 4.39. Alright? That kinda makes sense. That's the that's it for this one. Thanks for watching, and I'll see you in the next one.

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13. Non-Right Triangles

Law of Sines

Precalculus

13. Non-Right Triangles

Law of Sines

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Multiple Choice

Use the Law of Sines to find the length of side $a$ to two decimal places.

8.20

4.39

2.20

1.61

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Verified step by step guidance

Identify the given information: In triangle ABC, angle A is 45 degrees, angle B is 75 degrees, and side b is 6 units.

Recall the Law of Sines formula: $ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $. This formula relates the sides of a triangle to the sines of its angles.

Use the Law of Sines to set up the equation for side a: $ \frac{a}{\sin 60^\circ} = \frac{6}{\sin 75^\circ} $. Note that angle C can be found using the fact that the sum of angles in a triangle is 180 degrees, so angle C is 60 degrees.

Solve for side a: Rearrange the equation to solve for a, giving $ a = \frac{6 \cdot \sin 60^\circ}{\sin 75^\circ} $.

Calculate the value of a using the known values of $ \sin 60^\circ $ and $ \sin 75^\circ $ to find the length of side a to two decimal places.