1. Solve the inequality sin x + cos 2x > 1 if 0 ≤ x ≤ π/2
2. For a triangle ABC, it is given that : cos A + cos B + cos C = 3/2. Prove that the triangle is equilateral.
21Let sin x = t
⇒ cos 2x = 1 – 2t2
⇒ the inequality is : t + 1 – 2t2 > 1
⇒ t – 2t2 > 0
⇒ 2t2 – t < 0
⇒ t(2t – 1) < 0
⇒ (t – 0) (t – 1/2) < 0
⇒ 0 < t < 1/2
⇒ 0 < sin x < 1/2
In 0 ≤ x ≤ π/2, this means that 0 < x < π/6 is the solution
1708$\textbf{Here $\mathbf{A+B+C=\pi}$ and $\mathbf{0\leq A,B,C\leq \mathbf{\pi}}}$\\\\ Now Here Given that $\mathbf{cos\;A+\underbrace{\mathbf{cos\:B+cos\:C}}=\frac{3}{2}}$\\\\\\ $\mathbf{1-2sin^2\left(\frac{A}{2}\right)+2\cdot cos\left(\frac{B+C}{2}\right)\cdot cos\left(\frac{B-C}{2}\right)=\frac{3}{2}}$\\\\\\ $\mathbf{1-2\cdot sin^2\left(\frac{A}{2}\right)+2.sin\left(\frac{A}{2}\right).cos\left(\frac{B-C}{2}\right)=\frac{3}{2}}$\\\\\\ $\mathbf{4.sin^2\left(\frac{A}{2}\right)-4.sin\left(\frac{A}{2}\right).cos\left(\frac{B-C}{2}\right)+1=0}$\\\\\\ $\textbf{For Real Roots} \mathbf{Here\ D\geq 0}$\\\\ \textbf{So $\mathbf{16.cos^2\left(\frac{B-C}{2}\right)-4.4.1\geq 0}$}\\\\\\ So $\mathbf{cos^2\left(\frac{B-C}{2}\right)\geq 1\Leftrightarrow cos^2\left(\frac{B-C}{2}\right)=1\Leftrightarrow cos\left(\frac{B-C}{2}\right)=1}$\\\\\\ $\mathbf{\frac{B-C}{2}=0\Leftrightarrow B=C}$\\\\\\ Similarly We Get $\mathbf{A=C}$\\\\\\ So $\boxed{\boxed{\mathbf{A=B=C=\frac{\pi}{3}}}}$
21Jensen's inequality is very easy to grasp and solves lot's of problems.
search in TIIT and try to do the 2nd question .
1708yes Shubhodip you are saying Right. Using Jesan Inequality We can solve it with in 2 or 3 lines.
341actually what jagdish did is right. you cant use jensen for cosine in this domain as the function is not concave throughout
1708O Sorry bhatt Sir you are saying Right function f(x) = cos x is Concave down in [-pi / 2 , pi / 2].
So Sir How Can I solve Using Inequality in (ii) Question.
21but what i said was not wrong either :P
i have recalled how to do it by jensens
note that cosA + cosB ≤ 2 sinC2
so it follows that cosA + cosB + cosC ≤sinA2 + sinB2 + sinC2
By jensens sinA2 + sinB2 + sinC2 ≤32
probably i read that in Arihant's MO book
