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. Period of Sin\theta and Cos\theta is2\pi , whereas 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