Consider a triangle ABC for which ∠A=100∘,a=34,b=13. If such a triangle can not exist, then write NONE in each answer box. If there could be more than one such triangle, then enter dimensions for the one with the smallest value for side c. Finally, if there is a unique triangle ABC, then enter its dimensions.B is _______ degrees;∠C is_________ degrees;c= _________.

Answer:B is 22.12 degrees; ∠C is 57.88°; c=29.24Step-by-step explanation:So, first, it's important to draw a diagram of the triangle the problem is talking about (see  attached picture).Once the triangle has been drawn, we can visualize it better and determine what to do. So first, we are going to find what the value of angle B is by using law of sines:[tex]\frac{sin B}{b}=\frac{sin A}{a}[/tex]which can be solved for angle B:[tex]sin B=b\frac{sin A}{a}[/tex][tex] B= sin^{-1}(b\frac{sin A}{a})[/tex]and substitute the values we already know:[tex] B= sin^{-1}(13\frac{sin 100 ^{o}}{34})[/tex]which yields:B=22.12°Once we know what the angle of B is, we can now find the value of angle C by using the fact that the sum of the angles of any triangle is equal to 180°. So:A+B+C=180°When solving for C we get:C=180°-A-BC=180°-22.12°-|00°=57.88°So once we know what angle C is, we can go ahead and find the length of side c by using the law of sines again:[tex]\frac{c}{sin C}=\frac{a}{sin A}[/tex]and solve for c:[tex]c=sin C \frac{a}{sin A}[/tex]so we can now substitute for the values we already know:[tex]c=sin(57.88^{o})\frac{34}{sin(100^{o})}[/tex]which yields:c=29.24