Question

Could I get some Qualified Help on this Law of Sines question?

Solve the triangle.

A = 33, a = 19, b = 14

Answer 1

@Michele_Laino

Answer 2

If we apply the Law of Sines, we can write this equation:
\[\Large \frac{a}{{\sin A}} = \frac{b}{{\sin B}}\]

Answer 3

solving that equation for sinB, we get:
\[\Large \sin B = \frac{b}{a}\sin A\]

Answer 4

Could we reverse that and use \(\dfrac{sinA}a = \dfrac{sinB}b\)? Or is that not possible?

Answer 5

Right! It is possible

Answer 6

Awesome! I just find it easier when what I'm solving for is on top of the fraction :)

Answer 7

ok! :)

Answer 8

then substituting our numeric data, we can write:
\[\Large \sin B = \frac{b}{a}\sin A = \frac{{14}}{{19}} \times \sin 33\]

Answer 9

With that formula, I get \[\frac{ \sin(33) }{ 19 }=\frac{ \sin(B) }{ 14 }\]

Answer 10

that's right! @sleepyjess

Answer 11

Then, \[\frac{ 14\sin(33) }{ 19 } = 14\]

Answer 12

* = sin(B)

Answer 13

finally, making the right computation, we get:
\[\Large \sin B = \frac{b}{a}\sin A = \frac{{14}}{{19}} \times \sin 33 = 0.40131\]

Answer 14

So B is \[\approx \] 14!

Answer 15

Now since we now B and A, we can solve for C by 180-A-B=C?

Answer 16

*know

Answer 17

from which we can write the requested value of the amplitude B, as below:
\[\Large B \cong 23.7{\text{degrees}}\]

Answer 18

Oh wait, I messed up, B is about 23.7 degrees

Answer 19

Okay, so C is 180-33-23.7, so 123.3!

Answer 20

And then I use the same formula to find c.... so c is 29.2! :)

Answer 21

Now we can find the amplitude of the third angle, as below:
\[\Large C = 180 - A - B = 180 - 33 - 23.7 = 123.3\;{\text{degrees}}\]
That's right! @sleepyjess

Answer 22

Alternatively we can apply the Theorem of Carnot, in order to find the length of the side c, as below:
\[\Large {c^2} = {19^2} + {14^2} - 2 \times 19 \times 14\cos \left( {123.3} \right) \cong 849.08\]
and finally:
\[\Large c \cong 29.14\]