Question 15.2.26: Find the value of (i)sin 22⁰ 30′ (ii) cos 22⁰ 30′ (iii) tan......
Find the value of
(i) \sin 22^{\circ} 30^{\prime} (ii) \cos 22^{\circ} 30^{\prime} (iii) \tan 22^{\circ} 30^{\prime}
(i) \sin ^{2} \theta=\frac{(1-\cos 2 \theta)}{2} \\ \\\begin{array}{l}\Rightarrow \sin ^{2}\left(22^{\circ} 30^{\prime}\right)=\frac{\left(1-\cos 45^{\circ}\right)}{2}=\frac{\left(1-\frac{1}{\sqrt{2}}\right)}{2}=\frac{\sqrt{2}-1}{2 \sqrt{2}} \\ \\\Rightarrow \sin \left(22^{\circ} 30^{\prime}\right)=\sqrt{\frac{(\sqrt{2}-1)}{2 \sqrt{2}}} .\end{array}
(ii) \cos ^{2} \theta=\frac{(1+\cos 2 \theta)}{2} \\ \\\begin{array}{l}\Rightarrow \cos ^{2}\left(22^{\circ} 30^{\prime}\right)=\frac{\left(1+\cos 45^{\circ}\right)}{2}=\frac{\left(1+\frac{1}{\sqrt{2}}\right)}{2}=\frac{(\sqrt{2}+1)}{2 \sqrt{2}}\\ \\\Rightarrow \cos \left(22^{\circ} 30^{\prime}\right)=\sqrt{\frac{(\sqrt{2}+1)}{2 \sqrt{2}}} .\end{array}\\ \\
\begin{array}{l} \text{ (iii) }\tan ^{2}\left(22^{\circ} 30^{\prime}\right)=\frac{\sin ^{2}\left(22^{\circ} 30^{\prime}\right)}{\cos ^{2}\left(22^{\circ} 30^{\prime}\right)}=\frac{(\sqrt{2}-1)}{(2 \sqrt{2})} \times \frac{(2 \sqrt{2})}{(\sqrt{2}+1)}\\ \\ =\frac{\sqrt{2}-1}{\sqrt{2}+1}=\frac{(\sqrt{2}-1)}{(\sqrt{2}+1)} \times \frac{(\sqrt{2}-1)}{(\sqrt{2}-1)}\\ \\ =(\sqrt{2}-1)^{2} \\ \\ \Rightarrow \tan \left(22^{\circ} 30^{\prime}\right)=(\sqrt{2}-1) .\end{array}