| \[Angles→\] \[Ratios↓\] |
\[0^{\circ}\] | \[30^{\circ}\] | \[45^{\circ}\] | \[60^{\circ}\] | \[90^{\circ}\] |
|---|---|---|---|---|---|
| \[\sin\theta\] | \[0\] | \[\frac{1}{2}\] | \[\frac{1}{\sqrt{2}}\] | \[\frac{\sqrt{3}}{2}\] | \[1\] |
| \[\cos\theta\] | \[1\] | \[\frac{\sqrt{3}}{2}\] | \[\frac{1}{\sqrt{2}}\] | \[\frac{1}{2}\] | \[0\] |
| \[\tan\theta\] | \[0\] | \[\frac{1}{\sqrt{3}}\] | \[1\] | \[\sqrt{3}\] | \[Not \space Defined\] |
| \[\cosec\theta\] | \[Not \space Defined\] | \[2\] | \[\sqrt{2}\] | \[\frac{2}{\sqrt{3}}\] | \[1\] |
| \[\sec\theta\] | \[1\] | \[\frac{2}{\sqrt{3}}\] | \[\sqrt{2}\] | \[2\] | \[Not \space Defined\] |
| \[\cot\theta\] | \[Not \space Defined\] | \[\sqrt{3}\] | \[1\] | \[\frac{1}{\sqrt{3}}\] | \[0\] |
\[1) \space sin(7.5^{\circ})= \frac{\sqrt{2-\sqrt{2+\sqrt{3}}}}{2}= cos(82.5^{\circ})= sin \frac{\pi}{24}\]
\[2) \space cos(7.5^{\circ})= \frac{\sqrt{2+\sqrt{2+\sqrt{3}}}}{2}= sin(82.5^{\circ})= cos \frac{\pi}{24}\]
\[3) \space tan(7.5^{\circ})= \sqrt{6}-\sqrt{3}+\sqrt{2}-2=(\sqrt{2}-1)(\sqrt{3}-\sqrt{2})= cot(82.5^{\circ})= tan\frac{\pi}{24}\]
\[4) \space cot(7.5^{\circ})= \sqrt{6}+\sqrt{3}+\sqrt{2}+2=(\sqrt{2}+1)(\sqrt{3}+\sqrt{2})= tan(82.5^{\circ})= cot\frac{\pi}{24}\]
\[5) \space sin15^{\circ}= \frac{\sqrt{3}-1}{2\sqrt{2}}= cos75^{\circ}= sin \frac{\pi}{12}\]
\[6) \space cos15^{\circ}= \frac{\sqrt{3}+1}{2\sqrt{2}}= sin75^{\circ}= cos \frac{\pi}{12}\]
\[7) \space tan15^{\circ}= 2-\sqrt{3}= cot75^{\circ}= tan\frac{\pi}{12}\]
\[8) \space cot15^{\circ}= 2+\sqrt{3}= tan75^{\circ}= cot\frac{\pi}{12}\]
\[9) \space sin18^{\circ}= \frac{\sqrt{5}-1}{4}= \sqrt{\frac{3-\sqrt{5}}{8}} = cos72^{\circ}= sin\frac{\pi}{10} \]
\[10) \space cos18^{\circ}= \frac{\sqrt{10+2\sqrt{5}}}{4}= \sqrt{\frac{5+\sqrt{5}}{8}} = sin72^{\circ}= cos\frac{\pi}{10} \]
\[11) \space tan18^{\circ}= \sqrt{1-\frac{2\sqrt{5}}{5}}= cot72^{\circ}= tan\frac{\pi}{10}\]
\[12) \space cot18^{\circ}= \sqrt{5+2\sqrt{5}}= tan72^{\circ}= cot\frac{\pi}{10}\]
\[13) \space sin(22.5^{\circ})= \frac{\sqrt{2-\sqrt{2}}}{2}= \sqrt{\frac{4-\sqrt{8}}{8}} = cos(67.5^{\circ})= sin\frac{\pi}{8}\]
\[14) \space cos(22.5^{\circ})= \frac{\sqrt{2+\sqrt{2}}}{2}= \sqrt{\frac{4+\sqrt{8}}{8}} = sin(67.5^{\circ})= cos\frac{\pi}{8}\]
\[15) \space tan(22.5^{\circ})=\sqrt{2}-1=cot(67.5^{\circ})= tan\frac{\pi}{8}\]
\[16) \space cot(22.5^{\circ})=1+\sqrt{2}=tan(67.5^{\circ})= cot\frac{\pi}{8}\]
\[17) \space sin36^{\circ}= \frac{\sqrt{10-2\sqrt{5}}}{4}= \sqrt{\frac{5-\sqrt{5}}{8}}= cos54^{\circ}= sin\frac{\pi}{5}\]
\[18) \space cos36^{\circ}= \frac{\sqrt{5}+1}{4}= \sqrt{\frac{3+\sqrt{5}}{8}}= sin54^{\circ}=cos \frac{\pi}{5}\]
\[19) \space tan36^{\circ}= \sqrt{5-2\sqrt{5}}= cot54^{\circ}= tan \frac{\pi}{5}\]
\[20) \space cot36^{\circ}= \sqrt{\frac{5+2\sqrt{5}}{5}}= tan54^{\circ}= cot \frac{\pi}{5}\]
\[21) \space sin(37.5^{\circ})= \frac{\sqrt{2-\sqrt{2-\sqrt{3}}}}{2}= cos(52.5^{\circ})= sin \frac{5\pi}{24}\]
\[22) \space cos(37.5^{\circ})= \frac{\sqrt{2+\sqrt{2-\sqrt{3}}}}{2}= sin(52.5^{\circ})= cos \frac{5\pi}{24}\]
\[23) \space tan(37.5^{\circ})= \sqrt{6}+\sqrt{3}-\sqrt{2}-2=(\sqrt{2}+1)(\sqrt{3}-\sqrt{2})= cot(52.5^{\circ})= tan\frac{5\pi}{24}\]
\[24) \space cot(37.5^{\circ})= \sqrt{6}-\sqrt{3}-\sqrt{2}+2=(\sqrt{2}-1)(\sqrt{3}+\sqrt{2})= tan(52.5^{\circ})= cot\frac{5\pi}{24} \]