How do you solve a triangle given A = 58 degrees, a = 11.4, b = 12.8?

Hint:In the given question, we have been asked to find all the angles of a triangle and it is given that A = 58 degrees, a = 11.4, b = 12.8. In order to solve the question, first we need to apply the law of sines. We will get the measure of B in degrees and then later by angle sum property of the triangle we will find the measure of C in degrees.

Complete step by step solution:
We have given that,
\[A={{58}^{0}}\]
\[a=11.4\]
\[b=12.8\]
Applying the law of sines,
\[\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}\]
Substitute \[A={{58}^{0}},a=11.4,b=12.8\] in the above equation, we get
\[\Rightarrow \dfrac{\sin B}{12.8}=\dfrac{\sin {{\left( 58 \right)}^{0}}}{11.4}\]
Simplifying the above, we get
\[\Rightarrow \dfrac{1}{12.8}\times \sin B=\dfrac{\sin {{\left( 58 \right)}^{0}}}{11.4}\]
\[\Rightarrow 0.078\sin B=\dfrac{\sin {{\left( 58 \right)}^{0}}}{11.4}\]
By using the calculation, sin (58) degrees = 0.8480
Putting the value of \[\sin {{\left( 58 \right)}^{0}}\]in the above equation, we get
\[\Rightarrow 0.078\sin B=\dfrac{0.8480}{11.4}\]
Simplifying the above, we get
\[\Rightarrow 0.078\sin B=0.074\]
Solving the value of sin (B), we get
\[\Rightarrow \sin B=0.9521\]
Taking sin inverse on both the side of the equation, we get
\[\Rightarrow B={{\sin }^{-1}}\left( 0.9521 \right)\]
Evaluating the value of \[{{\sin }^{-1}}\left( 0.9521 \right)=72.2121\], we get
Therefore
\[\Rightarrow B=72.2121\]
\[\Rightarrow B={{\left( 72.2 \right)}^{0}}\]
As we know that sine function is positive in the first and second quadrant.
In the first quadrant, \[B={{\left( 72.2 \right)}^{0}}\]
And in the second quadrant,
\[\Rightarrow B={{\left( 180 \right)}^{0}}-{{\left( 72.2 \right)}^{0}}={{\left( 107.8 \right)}^{0}}\]
We will get two values of B.
To find the value of ‘c’,
\[\Rightarrow \dfrac{\sin {{\left( 14.2 \right)}^{0}}}{c}=\dfrac{\sin {{\left( 58 \right)}^{0}}}{11.4}\]
By putting all the values from above, we get
\[\Rightarrow c=3.3003\]
Therefore,
First triangle is of combination;
\[A={{58}^{0}}\]
\[B={{\left( 72.2 \right)}^{0}}\]
\[C={{180}^{0}}-{{58}^{0}}-{{\left( 72.2 \right)}^{0}}={{49.8}^{0}}\]
\[a=11.4\]
\[b=12.8\]
\[c=3.3003\]
Second triangle is of combination;
\[A={{58}^{0}}\]
\[B={{\left( 107.8 \right)}^{0}}\]
\[C={{180}^{0}}-{{58}^{0}}-{{107.8}^{0}}={{14.2}^{0}}\]
\[a=11.4\]
\[b=12.8\]
\[c=3.3003\]

Note: While solving these types of questions, students need to remember the law of sines. Law of sines is the proportionality of sides and angles of a triangle. This law of sine states that for the angles of a non-right angle triangle, each angle of the given triangle has the same ratio corresponding the angle measure to sine value such that \[\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}\].