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