Angles→Angles→
Ratios↓Ratios↓		 0∘0^{\circ} 30∘30^{\circ} 45∘45^{\circ} 60∘60^{\circ} 90∘90^{\circ}
sin⁡θ\sin\theta 00 12\frac{1}{2} 12\frac{1}{\sqrt{2}} 32\frac{\sqrt{3}}{2} 11
cos⁡θ\cos\theta 11 32\frac{\sqrt{3}}{2} 12\frac{1}{\sqrt{2}} 12\frac{1}{2} 00
tan⁡θ\tan\theta 00 13\frac{1}{\sqrt{3}} 11 3\sqrt{3} Not DefinedNot \space Defined
cosec⁡θ\cosec\theta Not DefinedNot \space Defined 22 2\sqrt{2} 23\frac{2}{\sqrt{3}} 11
sec⁡θ\sec\theta 11 23\frac{2}{\sqrt{3}} 2\sqrt{2} 22 Not DefinedNot \space Defined
cot⁡θ\cot\theta Not DefinedNot \space Defined 3\sqrt{3} 11 13\frac{1}{\sqrt{3}} 00

Values of some T−Ratios for many anglesValues \space of \space some \space T-Ratios \space for \space many \space angles

1) sin(7.5∘)=2−2+32=cos(82.5∘)=sinπ241) \space sin(7.5^{\circ})= \frac{\sqrt{2-\sqrt{2+\sqrt{3}}}}{2}= cos(82.5^{\circ})= sin \frac{\pi}{24}

2) cos(7.5∘)=2+2+32=sin(82.5∘)=cosπ242) \space cos(7.5^{\circ})= \frac{\sqrt{2+\sqrt{2+\sqrt{3}}}}{2}= sin(82.5^{\circ})= cos \frac{\pi}{24}

3) tan(7.5∘)=6−3+2−2=(2−1)(3−2)=cot(82.5∘)=tanπ243) \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) cot(7.5∘)=6+3+2+2=(2+1)(3+2)=tan(82.5∘)=cotπ244) \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) sin15∘=3−122=cos75∘=sinπ125) \space sin15^{\circ}= \frac{\sqrt{3}-1}{2\sqrt{2}}= cos75^{\circ}= sin \frac{\pi}{12}

6) cos15∘=3+122=sin75∘=cosπ126) \space cos15^{\circ}= \frac{\sqrt{3}+1}{2\sqrt{2}}= sin75^{\circ}= cos \frac{\pi}{12}

7) tan15∘=2−3=cot75∘=tanπ127) \space tan15^{\circ}= 2-\sqrt{3}= cot75^{\circ}= tan\frac{\pi}{12}

8) cot15∘=2+3=tan75∘=cotπ128) \space cot15^{\circ}= 2+\sqrt{3}= tan75^{\circ}= cot\frac{\pi}{12}

9) sin18∘=5−14=3−58=cos72∘=sinπ109) \space sin18^{\circ}= \frac{\sqrt{5}-1}{4}= \sqrt{\frac{3-\sqrt{5}}{8}} = cos72^{\circ}= sin\frac{\pi}{10}

10) cos18∘=10+254=5+58=sin72∘=cosπ1010) \space cos18^{\circ}= \frac{\sqrt{10+2\sqrt{5}}}{4}= \sqrt{\frac{5+\sqrt{5}}{8}} = sin72^{\circ}= cos\frac{\pi}{10}

11) tan18∘=1−255=cot72∘=tanπ1011) \space tan18^{\circ}= \sqrt{1-\frac{2\sqrt{5}}{5}}= cot72^{\circ}= tan\frac{\pi}{10}

12) cot18∘=5+25=tan72∘=cotπ1012) \space cot18^{\circ}= \sqrt{5+2\sqrt{5}}= tan72^{\circ}= cot\frac{\pi}{10}

13) sin(22.5∘)=2−22=4−88=cos(67.5∘)=sinπ813) \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) cos(22.5∘)=2+22=4+88=sin(67.5∘)=cosπ814) \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) tan(22.5∘)=2−1=cot(67.5∘)=tanπ815) \space tan(22.5^{\circ})=\sqrt{2}-1=cot(67.5^{\circ})= tan\frac{\pi}{8}

16) cot(22.5∘)=1+2=tan(67.5∘)=cotπ816) \space cot(22.5^{\circ})=1+\sqrt{2}=tan(67.5^{\circ})= cot\frac{\pi}{8}

17) sin36∘=10−254=5−58=cos54∘=sinπ517) \space sin36^{\circ}= \frac{\sqrt{10-2\sqrt{5}}}{4}= \sqrt{\frac{5-\sqrt{5}}{8}}= cos54^{\circ}= sin\frac{\pi}{5}

18) cos36∘=5+14=3+58=sin54∘=cosπ518) \space cos36^{\circ}= \frac{\sqrt{5}+1}{4}= \sqrt{\frac{3+\sqrt{5}}{8}}= sin54^{\circ}=cos \frac{\pi}{5}

19) tan36∘=5−25=cot54∘=tanπ519) \space tan36^{\circ}= \sqrt{5-2\sqrt{5}}= cot54^{\circ}= tan \frac{\pi}{5}

20) cot36∘=5+255=tan54∘=cotπ520) \space cot36^{\circ}= \sqrt{\frac{5+2\sqrt{5}}{5}}= tan54^{\circ}= cot \frac{\pi}{5}

21) sin(37.5∘)=2−2−32=cos(52.5∘)=sin5π2421) \space sin(37.5^{\circ})= \frac{\sqrt{2-\sqrt{2-\sqrt{3}}}}{2}= cos(52.5^{\circ})= sin \frac{5\pi}{24}

22) cos(37.5∘)=2+2−32=sin(52.5∘)=cos5π2422) \space cos(37.5^{\circ})= \frac{\sqrt{2+\sqrt{2-\sqrt{3}}}}{2}= sin(52.5^{\circ})= cos \frac{5\pi}{24}

23) tan(37.5∘)=6+3−2−2=(2+1)(3−2)=cot(52.5∘)=tan5π2423) \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) cot(37.5∘)=6−3−2+2=(2−1)(3+2)=tan(52.5∘)=cot5π2424) \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}