Use the Law of Sines to find the angle B B B to the nearest tenth of a degree .

Welcome back, everyone. Let's take a look at this practice problem. So we're used to using the law of sines to calculate missing sides in a non right triangle. But in this problem, we're gonna use the law of sines to calculate a missing angle. We're going to look for capital B in this non right triangle instead of one of the missing side lengths. We're going to round to the nearest tenth of a degree. Alright? Now we're going to start this the exact same way we start other law of sines problems, just by writing out the whole equation. Sine of angle a over a equals sine of angle b over b equals sine of angle c over c. Take a look at all the variables that you have. I've got angle a. That's big a over here. I don't know what b is. It's actually kind of what I'm looking for. But I and I also don't have what c is, so I don't know what that is. But I do have all of the lowercase letters. So I have what little a is, little b, and also little c. Alright? So if I'm trying to solve for this uppercase b over here, remember, I have to pick 2 out of these 3 ratios, which I know 3 out of 4 variables. If I try to pick this last these last two over here, I only know 2 out of the 4 variables, so I can't use that. I can't use these the the first and the third one because that doesn't include my target variable. I'm solving for b. So that basically kinda just narrows it down to really just one option here. You have to take a look at these first two terms because you have 3 out of 4 variables, and you're looking for angle b. Alright? So, basically, what I'm gonna do here is I'm just gonna move this b over to the other side, and I'm gonna rewrite this so that the sine of b is equal to lowercase b times this is the sine of angle a over a. Alright? Now it's just time to plug in some numbers because we know all of these numbers here. So we have the the sine of b is equal to little b, which is 6 times the sine of angle a, which is 30 degrees, divided by, little a, which is 4. There's a couple of ways to simplify this, but if you just plug all of this into your calculator, you should get 0.75. Alright? So remember, we're not quite done yet because this is the sine of b is equal to 0.75. I actually have to find what that angle is. If I have the sine of b, how do I get the angle? I just take the inverse sine. So in other words, this is b here is gonna be the inverse sine of 0.75. If you plug this in your calculator in degrees mode, what you should get is 48.6 degrees. Alright? So 48.6 degrees is our angle b. So if you go back to the diagram, this actually kind of makes sense because this looks like 30 degrees over here for a, and this angle over here kind of looks to be about 48.6 degrees. Now you may have been wondering that's isn't there another angle that when I take the sine of it, I get 0.75? And, yes, you are correct. There is another angle from 0 to 180 that gets you 0.75, but it's gonna be a really big number, like a 140 degrees. And And the reason that can't exist in this triangle is because if this is a 140 degrees, then these angles in this triangle wouldn't add up to a 180 because just this one and this one alone would be almost like a 170. Alright? So that's a good catch, but it's really just the sort of smaller number that you get when you take the inverse sine, 40 48.6 degrees. Alright? That's it for this one, folks. Let me know if you have any questions.

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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 angle $B$ to the nearest tenth of a degree.

48.6°

77.2°

40.5°

35.3°

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Verified step by step guidance

Identify the given values in the triangle: side a = 4, side b = 6, side c = 7.8, and angle C = 30 degrees.

Recall the Law of Sines formula: $ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $.

Use the Law of Sines to find angle B. Start by setting up the equation: $ \frac{b}{\sin B} = \frac{c}{\sin C} $. Substitute the known values: $ \frac{6}{\sin B} = \frac{7.8}{\sin 30^\circ} $.

Calculate $ \sin 30^\circ $, which is 0.5, and substitute it into the equation: $ \frac{6}{\sin B} = \frac{7.8}{0.5} $.

Solve for $ \sin B $ by cross-multiplying and isolating $ \sin B $: $ \sin B = \frac{6 \times 0.5}{7.8} $. Then, use the inverse sine function to find angle B to the nearest tenth of a degree.