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