Mathematics - Functions Question with Solution | TestHub

Given $\mathrm{f}$ be an odd function $f(x)=3\sin x+4\cos x$ Now, $$\begin{array}{r}f\left(\frac{-11\pi }{6}\right)=3\sin \left(\frac{-11\pi }{6}\right)+4\cos \left(\frac{-11\pi }{6}\right)\\ f\left(\frac{-11\pi }{6}\right)=3\sin \left(-2\pi +\frac{\pi }{6}\right)+4\cos \left(-2\pi +\frac{\pi }{6}\right)\\ f\left(\frac{-11\pi }{6}\right)=3\sin \left\{-\left(2\pi -\frac{\pi }{6}\right)\right\}+4\cos \left\{-\left(2\pi -\frac{\pi }{6}\right)\right\}\\ \left\{\text{ For odd functions }\begin{array}{l}\sin (-\theta )=-\sin \theta \\ \text{ and }\cos (-\theta )=-\cos \theta \end{array}\right\}\end{array}$$ $$\begin{array}{rl}\therefore f\left(\frac{-11\pi }{6}\right)=-3\sin \left(2\pi -\frac{\pi }{6}\right)-4\cos \left(2\pi -\frac{\pi }{6}\right)& \\ \Rightarrow f\left(\frac{-11\pi }{6}\right)& =+3\sin \left(\frac{\pi }{6}\right)-4\cos \frac{\pi }{6}\\ \Rightarrow f\left(\frac{-11\pi }{6}\right)& =3\times \frac{1}{2}-4\times \frac{\sqrt{3}}{2}\\ \text{ or }f\left(\frac{-11\pi }{6}\right)& =\frac{3}{2}-2\sqrt{3}\end{array}$$