Identitas Trigonometri

Matematika adalah bahasa dari fisika. Jadi terkadang mengerjakan soal fisika ketemu aturan hitungan di matematika. Ada beberapa aturan perhitungan di matematika yang dipakai di fisika misal logaritma, turunan, integral, trigonometri dan masih banyak lagi. Pada laman ini saya sajikan aturan identitas  trigonometeri yang dipakai di fisika.

Hafalin dulu tabel trigonometeri

Kuadran I, II, III, dan IV

Identitas trigonometri yang sering dipakai di Fisika.

01. $\displaystyle \frac {1}{sin\; \alpha}=\;cosec\; \alpha$

02. $\displaystyle \frac {1}{cos\; \alpha}=\;sec\; \alpha$

03. $\displaystyle \frac {sin\; \alpha}{cos\; \alpha}=\; tan\; \alpha$

04. $\displaystyle \frac {cos\; \alpha}{sin\; \alpha}=\; cotan\; \alpha$

05. $\displaystyle \frac {1}{tan\; \alpha}=\; cotan\; \alpha$

06. $\displaystyle sin^{2}\alpha + cos^{2}\alpha =\; 1$

07. $\displaystyle sec^{2}\alpha =1+ tan^{2}\alpha$

08. $\displaystyle cosec^{2}\alpha =1+ cotan^{2}\alpha =\; 1$

09. $\displaystyle sin\;2\alpha=\;2sin\;\alpha\;cos\;\alpha$

10. $\displaystyle cos\;2\alpha=\;2cos^{2}\alpha - 1=\;1-2sin^{2}\alpha$

11. $\displaystyle 2cos\;\alpha\;sin\;\beta=\;sin(\alpha+\beta)-sin(\alpha-\beta)$

12. $\displaystyle 2cos\alpha\;cos\beta=\;sin(\alpha+\beta)+cos(\alpha-\beta)$

13. $\displaystyle 2sin\;\alpha\;cos\;\beta=\;sin(\alpha+\beta)+sin(\alpha-\beta)$

14. $\displaystyle -2cos\;\alpha\;cos\;\beta=\;cos(\alpha+\beta)-cos(\alpha-\beta)$

15. $\displaystyle sin^{2}\alpha=\;\frac {1}{2}-\frac {1}{2}cos\;2\alpha$

16. $\displaystyle cos^{2}\alpha=\;\frac{1}{2}+\frac{1}{2}cos\;2\alpha$

17. $\displaystyle sin\alpha +cos\;\beta=\;2 sin\frac{1}{2}(\alpha+\beta)\;cos\frac{1}{2}(\alpha+\beta)$

18. $\displaystyle sin\alpha -cos\;\beta=\;2 cos\frac{1}{2}(\alpha+\beta)\;sin\frac{1}{2}(\alpha+\beta)$

19. $\displaystyle cos\alpha +cos\;\beta=\;2 cos\frac{1}{2}(\alpha+\beta)\;sin\frac{1}{2}(\alpha+\beta)$

20.  $\displaystyle cos\alpha -cos\;\beta=\;-2 cos\frac{1}{2}(\alpha+\beta)\;sin\frac{1}{2}(\alpha+\beta)$

21.  $\displaystyle tan (\alpha +\beta)=\frac {tan\;\alpha +tan\;\beta}{1-tan\;\alpha.tan\;\beta}$

22. $\displaystyle tan (\alpha - \beta)=\frac {tan\;\alpha -tan\;\beta}{1+tan\;\alpha.tan\;\beta}$

Sudut Rangkap

• $\displaystyle sin\;2\alpha=\;2sin\;\alpha.cos\;\alpha$
• $\displaystyle cos\;2\alpha=\;cos^{2}\alpha - sin^{2}\alpha$
• $\displaystyle cos\;2\alpha=\;cos^{2}\alpha-1$
• $\displaystyle cos\;2\alpha=\;1-sin^{2}\alpha$
• $\displaystyle tan\;2\alpha=\;\frac{2tan\;\alpha}{1-tan^{2}\alpha}$

Jumlah dan Selisih Dua Sudut

• $\displaystyle sin(A+B)=\;sin A.cos B+cos A. sin B$
• $\displaystyle sin(A-B)=\;sin A.cos B-cos A. sin B$
• $\displaystyle cos(A+B)=\;cosA.cos B-sin A. sin B$
• $\displaystyle cos(A-B)=\;cosA.cos B+sin A. sin B$
• $\displaystyle tan (A+B)=\frac{tanA+tanB}{1-tanA.tanB}$
• $\displaystyle tan (A-B)=\frac{tanA-tanB}{1+tanA.tanB}$

Sudut Tengahan

• $\displaystyle cos\left (\frac {\theta}{2} \right)=\pm\sqrt{\frac {1+cos\;\theta}{2}}$
• $\displaystyle sin\left (\frac {\theta}{2} \right)=\pm\sqrt{\frac {1-cos\;\theta}{2}}$
• $\displaystyle tan\left (\frac {\theta}{2} \right)=\pm\sqrt{\frac {1-cos\;\theta}{1+cos\;\theta}}$
• $\displaystyle tan\left (\frac {\theta}{2} \right)=\frac {sin\;\theta}{1+cos\;\theta}$
• $\displaystyle tan\left (\frac {\theta}{2} \right)=\frac {1-cos\;\theta}{sin\;\theta}$