Answer:
All sides of the triangle are a = 34, b = 27 and c = 46.7
and angles are A = 46°, B = 34.83° and C = 99.17°
Step-by-step explanation:
In a given triangle A = 46°, a = 34 units and b = 27 units
Then we have to find all angles and measure of the side left.
By sine rule,
![\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}](https://tex.z-dn.net/?f=%5Cfrac%7Ba%7D%7BsinA%7D%3D%5Cfrac%7Bb%7D%7BsinB%7D%3D%5Cfrac%7Bc%7D%7BsinC%7D)
![\frac{a}{sinA}=\frac{b}{sinB}](https://tex.z-dn.net/?f=%5Cfrac%7Ba%7D%7BsinA%7D%3D%5Cfrac%7Bb%7D%7BsinB%7D)
![\frac{34}{sin46}=\frac{27}{sinB}](https://tex.z-dn.net/?f=%5Cfrac%7B34%7D%7Bsin46%7D%3D%5Cfrac%7B27%7D%7BsinB%7D)
sinB = ![\frac{27\times sin46}{34}](https://tex.z-dn.net/?f=%5Cfrac%7B27%5Ctimes%20sin46%7D%7B34%7D)
sinB = 0.5712
B = ![sin^{-1}(0.5712)](https://tex.z-dn.net/?f=sin%5E%7B-1%7D%280.5712%29)
B = 34.83°
Since in a triangle,
∠A + ∠B + ∠C = 180°
46°+ 34.83° + ∠C = 180°
80.83° + ∠C = 180°
∠C = 180 - 80.83 = 99.17°
![\frac{b}{sinB}=\frac{c}{sinC}](https://tex.z-dn.net/?f=%5Cfrac%7Bb%7D%7BsinB%7D%3D%5Cfrac%7Bc%7D%7BsinC%7D)
![\frac{27}{sin34.83}=\frac{c}{sin99.17}](https://tex.z-dn.net/?f=%5Cfrac%7B27%7D%7Bsin34.83%7D%3D%5Cfrac%7Bc%7D%7Bsin99.17%7D)
![\frac{27}{0.5711}=\frac{c}{0.9872}](https://tex.z-dn.net/?f=%5Cfrac%7B27%7D%7B0.5711%7D%3D%5Cfrac%7Bc%7D%7B0.9872%7D)
c = ![\frac{27\times 0.9872}{0.5711}=46.67](https://tex.z-dn.net/?f=%5Cfrac%7B27%5Ctimes%200.9872%7D%7B0.5711%7D%3D46.67)
Therefore, all sides of the triangle are a = 34, b = 27 and c = 46.7
and angles are A = 46°, B = 34.83° and C = 99.17°