Question

Please solve this step by step with proper reason. DO NOT SPAM.​

Answer 1

Answer:

$B \: = \: \dfrac{\pi}{2}$

Step-by-step explanation:

Given ----> Α = π / 6 and b : c = 2 :√3

To find ----->

$value \: of \: angle \: b \:$

Concept used ---->

1) Cos rule

CosA = ( b² + c² - a² ) / 2 bc

2) Sine rule

a / sinA = b / sinB = c / sinC

Solution---> ATQ,

$b : \: c \: = 2 : \: \sqrt{3}$

$let \: b \: = \: 2k \: \: and \: c \: = \: \sqrt{3}k$

$now \\ cosA \: = \dfrac{ {b}^{2} \: + \: {c}^{2} \: - \: {a}^{2} }{2 \: b \: c}$

$cos \dfrac{\pi}{6} \: = \dfrac{( \: 2k \: ) ^{2} \: + ( \: \sqrt{3 \: k \:} \: ) ^{2} \: - \: {a}^{2} }{2 \: ( \: 2k \: ) \: ( \: \sqrt{3} \: k \: ) }$

$= > \dfrac{ \sqrt{3} }{2} \: = \: \dfrac{4 {k}^{2} \: + \: 3 {k}^{2} \: - \: {a}^{2} }{4 \: \sqrt{3} \: {k}^{2} }$

$= > 12 \: {k}^{2} \: = \: 2 \: ( \: 7 {k}^{2} \: - {a}^{2} \: )$

$= > 6 \: {k}^{2} \: = 7 \: {k}^{2} \: - \: {a}^{2}$

$= > {a}^{2} \: = \: 7 {k}^{2} \: - \: 6 {k}^{2}$

$= > {a}^{2} \: = \: {k}^{2}$

$= > \: a \: = \: k$

$now \: by \: sine \: rule \: we \: get \\ \dfrac{a}{sinA} \: = \dfrac{b}{sinB}$

$= > \dfrac{k}{sin \frac{\pi}{6} } \: = \: \dfrac{2k}{sinB}$

$= > \: k \: sinB \: = \: 2k \: sin \dfrac{\pi}{6}$

$= > k \: cancel \: out \: from \: both \: sides$

$= > \: sinB \: = \:2 sin \dfrac{\pi}{6}$

$= > \: sinB \: \: = \: 2 \: \dfrac{1}{2}$

$= > \: sinB \: = \: 1 \\ = > \: sinB \: = \: sin \frac{\pi}{2 } \\ = > \: B \: = \: \frac{\pi}{2}$