Question

The angles $A, B$ and $C$ of a triangle $ABC$ are in A.P. If $b : c =\sqrt {3} : \sqrt {2}$ then the angle $A$ is

• A. 30$^\circ$
• B. 15$^\circ$
• C. 75$^\circ$
• D. 45$^\circ$

Answer 1

The Correct Option is C

The correct answer is C:\(75\degree\)
Given data:
In \(\triangle{ABC}\)
\(b\ratio{c}=\sqrt{3}\ratio\sqrt{2}\)
we know that,
\(\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}\)-(i)
the angles A,B,C are in A.P(Given)
\(\therefore 2B=A+C\)
we know that, \(\angle{A}+\angle{B}+\angle{C}=180\degree\)
⇒\(2\angle{B}+\angle{B}=180\degree\)
⇒\(\angle{B}=60\degree\)
From equation (i),  \(\frac{b}{c}=\frac{sinB}{sin{C}}\)
⇒\(\frac{\sqrt{3}}{\sqrt{2}}=\frac{sin60\degree}{sinc}\)
⇒\(\frac{\sqrt{3}}{\sqrt{2}}=\frac{\frac{\sqrt{3}}{2}}{sinc}\)
⇒\(sinc=\frac{1}{\sqrt{2}}\)
⇒\(\angle{C}=45\degree\)
\(\therefore \angle{A}+\angle{B}+\angle{C}=180\degree\)
⇒\(\angle{A}=180\degree-60\degree-45\degree\)
⇒\(A=75\degree\)