A) \[\sin \,\,{{15}^{o}}\]
B) \[\cos \,\,{{15}^{o}}\]
C) \[sin\,{{15}^{o}}.\,\cos \,{{15}^{o}}\]
D) \[sin\,{{15}^{o}}.\,\cos \,{{75}^{o}}\]
Correct Answer: C
Solution :
[a] \[\sin {{15}^{o}}=\sin ({{45}^{o}}-{{30}^{o}})\] \[=\sin {{45}^{o}}.\,\,\cos \,{{30}^{o}}-\cos \,{{45}^{o}}.\,\sin {{30}^{o}}\] \[=\frac{\sqrt{3}}{2\sqrt{2}}-\frac{1}{2\sqrt{2}}=\left( \frac{\sqrt{3}-1}{2\sqrt{2}} \right)\] [b] \[\cos \,{{15}^{o}}=\cos \,({{45}^{o}}-{{30}^{o}})\] \[=\cos \,{{45}^{o}}.\cos {{30}^{o}}-\cos {{45}^{o}}.\sin {{30}^{o}}\] \[=\frac{\sqrt{3}}{2\sqrt{2}}+\frac{1}{2\sqrt{2}}=\left( \frac{\sqrt{3}+1}{2\sqrt{2}} \right)\] [c] \[\sin {{15}^{o}}.\cos {{15}^{o}}=\frac{1}{2}\,(2\sin {{15}^{o}}.\,\cos \,{{15}^{o}})\] \[=\frac{1}{2}\,\sin \,{{30}^{o}}=\frac{1}{4}\] [d] \[\sin {{15}^{o}}.\cos {{75}^{o}}=\sin {{15}^{o}}.\cos ({{90}^{o}}-{{15}^{o}})\] \[=\sin {{15}^{o}}.\sin {{15}^{o}}\] \[={{\sin }^{2}}\,{{15}^{o}}={{\left( \frac{\sqrt{3}-1}{2\sqrt{2}} \right)}^{2}}\] \[=\frac{3+1-2\sqrt{3}}{4.24}\,=\left( \frac{2-\sqrt{3}}{4} \right)\]
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