# Results of Trigonometric Ratios of Allied Angles

 $\alpha$ $Sin\alpha$ $Cos\alpha$ $Tan\alpha$ $Cot\alpha$ $Sec\alpha$ $Co\sec \alpha$ $- \theta$ $- Sin\theta$ $+ Cos\theta$ $- Tan\theta$ $- Cot\theta$ $+ Sec\theta$ $- Co\sec \theta$ ${90^ \circ } - \theta$ $+ Cos\theta$ $+ Sin\theta$ $+ Cot\theta$ $+ Tan\theta$ $+ Co\sec \theta$ $+ Sec\theta$ ${90^ \circ } + \theta$ $+ Cos\theta$ $- Sin\theta$ $- Cot\theta$ $- Tan\theta$ $- Co\sec \theta$ $+ Sec\theta$ ${180^ \circ } - \theta$ $+ Sin\theta$ $- Cos\theta$ $- Tan\theta$ $- Cot\theta$ $- Sec\theta$ $+ Co\sec \theta$ ${180^ \circ } + \theta$ $- Sin\theta$ $- Cos\theta$ $+ Tan\theta$ $+ Cot\theta$ $- Sec\theta$ $- Co\sec \theta$ ${270^ \circ } - \theta$ $- Cos\theta$ $- Sin\theta$ $+ Cot\theta$ $+ Tan\theta$ $- Co\sec \theta$ $- Sec\theta$ ${270^ \circ } + \theta$ $- Cos\theta$ $+ Sin\theta$ $- Cot\theta$ $- Tan\theta$ $+ Co\sec \theta$ $- Sec\theta$ ${360^ \circ } - \theta$ $- Sin\theta$ $+ Cos\theta$ $- Tan\theta$ $- Cot\theta$ $+ Sec\theta$ $- Co\sec \theta$ ${360^ \circ } + \theta$ $+ Sin\theta$ $+ Cos\theta$ $+ Tan\theta$ $+ Cot\theta$ $+ Sec\theta$ $+ Co\sec \theta$

1. $Sin( - \theta ) = - Sin\theta$

2. $Cos( - \theta ) = Cos\theta$

3. $Tan( - \theta ) = - Tan\theta$

4. $Sin({90^ \circ } - \theta ) = Cos\theta$

5. $Cos({90^ \circ } - \theta ) = Sin\theta$

6. $Tan({90^ \circ } - \theta ) = Cot\theta$

7. $Sin({180^ \circ } - \theta ) = Sin\theta$

8. $Cos({180^ \circ } - \theta ) = - Cos\theta$

9. $Tan({180^ \circ } - \theta ) = - Tan\theta$

10. $Sin({270^ \circ } - \theta ) = - Cos\theta$

11. $Cos({270^ \circ } - \theta ) = - Sin\theta$

12. $Tan({270^ \circ } - \theta ) = Cot\theta$

13. $Sin({90^ \circ } + \theta ) = Cos\theta$

14. $Cos({90^ \circ } + \theta ) = - Sin\theta$

15. $Tan({90^ \circ } + \theta ) = - Cot\theta$

16. $Sin({180^ \circ } + \theta ) = - Sin\theta$

17. $Cos({180^ \circ } + \theta ) = - Cos\theta$

18. $Tan({180^ \circ } + \theta ) = Tan\theta$

19. $Sin({270^ \circ } + \theta ) = - Cos\theta$

20. $Cos({270^ \circ } + \theta ) = Sin\theta$

21. $Tan({270^ \circ } + \theta ) = - Cot\theta$

22. The period of $Sin\theta$ and $Cos\theta$ is $2\pi$, whereas the period of $Tan\theta$ and $Cot\theta$ is $\pi$.

If $k$ is any integer, then

23. $Sin(k\pi ) = 0$

24. $Cos(k\pi ) = {( - 1)^k}$

25. $Sin(k\pi + \beta ) = {( - 1)^k}Sin\beta$

26. $Cos(k\pi + \beta ) = {( - 1)^k}Cos\beta$

27. $Sin\left[ {(2k + 1)\frac{\pi }{2} + \beta } \right] = {( - 1)^k}Cos\beta$

28. $Cos\left[ {(2k + 1)\frac{\pi }{2} + \beta } \right] = {( - 1)^{k + 1}}Sin\beta$