IIT-JEE 2004 ScreeningQuestion 1
Given both $\theta$ and $\phi$ are acute angles and $\sin\theta=\dfrac12$, $\cos\phi=\dfrac13$, then the value of $\theta+\phi$ belongs to
Trigonometric Ratio & Identities
Solution
Concept. $\theta=\frac\pi6$ exactly; $\cos\varphi=\frac13$ places $\varphi$ between $\frac\pi3$ and $\frac\pi2$. Formulas used. (1) $\cos$ decreases on $\left[0,\frac\pi2\right]$, with $\cos\frac\pi3=\frac12$, $\cos\frac\pi2=0$Step 1 - $\theta$. $\sin\theta=\frac12$ and $\theta$ acute give $\theta=\frac\pi6$. Step 2 - $\varphi$. $0<\frac13<\frac12$, so by (1) $\frac\pi3<\varphi<\frac\pi2$. Step 3 - Add. $\frac\pi6+\frac\pi3<\theta+\varphi<\frac\pi6+\frac\pi2$, i.e. $\theta+\varphi\in\left(\frac\pi2,\frac{2\pi}{3}\right)$. Answer: (B)