Question

In this diagram, what is the value of b, correct to the nearest tenth
a)2.5
b)2.8
c)3.7
d)5.0
e)6.2

please help !

Answer 1

I think by "b", you mean the length of side opposite to "angle b"

Here is a relation

\(\huge{\frac{a}{sin \ a} = \frac{b}{sin \ b} =\frac{c}{sin \ c}}\)

where a,b,c are lengths of sides opposite to angles a,b,c

Answer 2

Take here side c=3 ,angle c=26.16
Then side b=? and angle b=54.33

Answer 3

First off to make this easy on us, let's write out what we have and what we don't have.

We have

We have Angle C=26. We have side c=3
We have Angle B=54.33.

We don't have

We don't have Angle A ____. We don't have side A_____
We don't have side B_____

Now remember the rule \[\frac{ a }{ \sin A }= \frac{ b }{ \sin B }=\frac{ c }{ \sin C }\]

Now let's plug in what we have \[\frac{ a }{ \sin A }= \frac{ b }{ \sin 54.33 }=\frac{ 3 }{ \sin 26.16 }\]

Now since we don't have \[\frac{ a }{\sin A }\] Let's leave it out for now. So let's go back to the two equation we have a number in and solve those first. \[\frac{ b }{ \sin 54.33 }=\frac{ 3 }{ \sin 26.16 }\]

Now let's get b alone and solve the problem. Let's use our algebraic skills to slove this problem. What we have to do first is multiply by \[\sin 54.33\] to both sides. And our problem comes out to be \[b =\frac{ 3 }{\sin 26.16 } \left( \sin 54.33 \right)\] So when you divide and multiply it out \[b=5.5\]

Now let's go back and solve for a. You might say now, AHH!!! We have nothing for A. How are we going to derive the angle or the side?!?!!? It's okay we got this. If you remember from previous algebra courses a triangle equals 180. So add the two angles we are given and subtract from 180. \[180-(26.16+54.33)= Angle A\]So your answer will come out to be 99.51. So\[Angle A=99.51\]

So you might say, okay I'm relaxed now but how will I find side a. Well you will find side A using the equation we used before \[\frac{ a }{ \sin A }= \frac{ b }{ \sin B }=\frac{ c }{ \sin C }\]

So now you could either equal a/sin A to b/sin B or c/sin C as you have both of them. It doesn't matter which one you use you still get the same thing. So in this case let's use c/sin C. So our equation will come out to be\[\frac{ a }{ \sin99.51 }=\frac{ 3 }{ \sin26.16 }\] Just like before use algebraic skills to solve it out. So multiply by sin 99.51 to get side a alone because we are solving for that. So the equation is now.\[a=\frac{ 3 }{ \sin26.16 } \left( \sin 99.51 \right)\] When you divide and multiply it out it becomes

\[a=0.11\]

Oh Wow! I didn't realize you said only b but this will help you solve for a if you are asked the next time. So b was 5.5 and this 5.5 is close to 5.0 than 6.2 the answer is D)5.0