Find the principal solutions of each of the following equations:

(i) $\sin x=\frac{1}{2} \quad$ (ii) $\cos x=\frac{1}{\sqrt{2}} \quad$ (iii) $\tan x=\frac{1}{\sqrt{3}}$

Question

Find the principal solutions of each of the following equations:

(i) [latex] \sin x=\frac{1}{2} \quad[/latex](ii) [latex] \cos x=\frac{1}{\sqrt{2}} \quad[/latex](iii) [latex] an x=\frac{1}{\sqrt{3}} [/latex]

Accepted Answer

(i) The given equation is [latex] \sin x=\frac{1}{2} [/latex].

We know that [latex] \sin \frac{\pi}{6}=\frac{1}{2} [/latex] and [latex] \sin \left(\pi-\frac{\pi}{6} ight)=\frac{1}{2} [/latex].

[latex] herefore \quad \sin \frac{\pi}{6}=\frac{1}{2} [/latex] and [latex] \sin \frac{5 \pi}{6}=\frac{1}{2} [/latex].

Hence, the principal solutions are [latex] x=\frac{\pi}{6} [/latex] and [latex] x=\frac{5 \pi}{6} [/latex].

(ii) The given equation is [latex] \cos x=\frac{1}{\sqrt{2}} [/latex].

We know that [latex] \cos \frac{\pi}{4}=\frac{1}{\sqrt{2}} [/latex] and [latex] \cos \left(2 \pi-\frac{\pi}{4} ight)=\frac{1}{\sqrt{2}} [/latex].

[latex] herefore \quad \cos \frac{\pi}{4}=\frac{1}{\sqrt{2}} [/latex] and [latex] \cos \frac{7 \pi}{4}=\frac{1}{\sqrt{2}} [/latex].

Hence, the principal solutions are [latex] x=\frac{\pi}{4} [/latex] and [latex] x=\frac{7 \pi}{4} [/latex].

(iii) The given equation is [latex] an x=\frac{1}{\sqrt{3}} [/latex].

We know that [latex] an \frac{\pi}{6}=\frac{1}{\sqrt{3}} [/latex] and [latex] an \left(\pi+\frac{\pi}{6} ight)=\frac{1}{\sqrt{3}} [/latex].

[latex] herefore an \frac{\pi}{6}=\frac{1}{\sqrt{3}} [/latex] and [latex] an \frac{7 \pi}{6}=\frac{1}{\sqrt{3}} [/latex].

Hence, the principal solutions are [latex] x=\frac{\pi}{6} [/latex] and [latex] x=\frac{7 \pi}{6} [/latex].

Question 17.2.1: Find the principal solutions of each of the following equati......