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Hubungan fungsi trigonometri
Fungsi dasar:
![{\displaystyle \sin A={\frac {a}{c}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b45d8020c93034babbf051ed6ae934bda523925)
![{\displaystyle \cos A={\frac {b}{c}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fff2735d9bc06ddca01538e5b6d4d00999002ac)
![{\displaystyle \tan A={\frac {\sin A}{\cos A}}\ ={\frac {a}{b}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfa43d47fdb68e77537282e43f6f749e4bce0276)
![{\displaystyle \cot A={\frac {1}{\tan A}}={\frac {\cos A}{\sin A}}\ ={\frac {b}{a}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81296f96d1db02699dada0f4b83a4f715e1c24fd)
![{\displaystyle \sec A={\frac {1}{\cos A}}\ ={\frac {c}{b}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79062e7f068102d5d77a0195076761ac0e01c013)
![{\displaystyle \csc A={\frac {1}{\sin A}}\ ={\frac {c}{a}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/021a1d1a11a569064588b76d7ef7281f02a6745a)
![{\displaystyle \sin {(-A)}=-\sin A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11de5bbafabf8fddd5e7cc86788c32eeac130056)
![{\displaystyle \cos {(-A)}=\cos A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf976b6b7ed005d80280ce80fd1b8dee1e9d1650)
![{\displaystyle \tan {(-A)}=-\tan A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7155d43332d43f9ed85b184ea9394753e7d288c0)
![{\displaystyle \csc {(-A)}=-\csc A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e872c0824afb895429db81b043966d92f84b18ec)
![{\displaystyle \sec {(-A)}=\sec A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efc7560ddfcce5991fd1e85002a7882c5fa1c065)
![{\displaystyle \cot {(-A)}=-\cot A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54ccf39271ec787042496f5bbe4d2b5ee0b75cd1)
Identitas trigonometri
![{\displaystyle \sin ^{2}A+\cos ^{2}A=1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ff50410076cba4dbba8fba1e65c1238b003860)
![{\displaystyle 1+\tan ^{2}A={\frac {1}{\cos ^{2}A}}=\sec ^{2}A\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaff909fe62b06b639d51914d9f0a33f4ff3139f)
![{\displaystyle 1+\cot ^{2}A={\frac {1}{\sin ^{2}A}}=\csc ^{2}A\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/679bc1ea255fed02a784860a5e089686068eb41e)
Penjumlahan
![{\displaystyle \sin(A+B)=\sin A\cos B+\cos A\sin B\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8940244b30faa3bacf1e99ddb3f66b5d3b1f01b)
![{\displaystyle \sin(A-B)=\sin A\cos B-\cos A\sin B\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91535667a245d188ace424a53ece21fa0ca463c8)
![{\displaystyle \cos(A+B)=\cos A\cos B-\sin A\sin B\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4d42a97f2a96f5197ff11f64198b0508da8f37)
![{\displaystyle \cos(A-B)=\cos A\cos B+\sin A\sin B\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f3b3a20893c660a0b564e620a3152af7294d8c)
![{\displaystyle \tan(A+B)={\frac {\tan A+\tan B}{1-\tan A\tan B}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c273cf6cb4037c56e67b4a9df09f416acf6f04f)
![{\displaystyle \tan(A-B)={\frac {\tan A-\tan B}{1+\tan A\tan B}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce2901384be2006d81fb3dd03210382b6cc1020)
![{\displaystyle sin(u)+sin(v)=2sin({\frac {u+v}{2}})cos({\frac {u-v}{2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a90995f2700b26d08dd969506e5df2b29bf78ae5)
![{\displaystyle sin(u)-sin(v)=2cos({\frac {u+v}{2}})sin({\frac {u-v}{2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6552133e07d72fe450b257c8d9ac5cfa9f3a015)
![{\displaystyle cos(u)+cos(v)=2cos({\frac {u+v}{2}})cos({\frac {u-v}{2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d56dbce6ec8fc0791cf7b2f5cb7a02e931cd166)
![{\displaystyle cos(u)-cos(v)=-2sin({\frac {u+v}{2}})sin({\frac {u-v}{2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4536bcb97cede04ed444fe2a12fd64425480c11)
Perkalian
![{\displaystyle 2\sin A\times \cos B=\sin(A+B)+\sin(A-B),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b540167c2fb939f59ac808313bdf0e8088c7266d)
![{\displaystyle 2\cos A\times \sin B=\sin(A+B)-\sin(A-B),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e7b0fd4139c4113397cc6ad5d8f033dcabbd75)
![{\displaystyle 2\cos A\times \cos B=\cos(A+B)+\cos(A-B),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0b3ee6aedd0860dd6d21aad2ce603089760f38)
![{\displaystyle 2\sin A\times \sin B=-\cos(A+B)+\cos(A-B),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/038faebfd46140451c88fa630d2290867d5f1432)
Rumus sudut rangkap dua
![{\displaystyle \sin 2A=2\sin A\cos A\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/947d2efa3a6ddf3bc9133e30195c148dc2c5c935)
![{\displaystyle \cos 2A=\cos ^{2}A-\sin ^{2}A=2\cos ^{2}A-1=1-2\sin ^{2}A\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1824e3723fa8998e13c9e0cc122b6961b18c0ed8)
![{\displaystyle \tan 2A={2\tan A \over 1-\tan ^{2}A}={2\cot A \over \cot ^{2}A-1}={2 \over \cot A-\tan A}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bcc5de0fe4cdd69def6d7c1515d2ed936c4cc07)
Rumus sudut rangkap tiga
![{\displaystyle \sin 3A=3\sin A-4\sin ^{3}A\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14c0d33545ef95a97cce4b448ca2028837bd9a1f)
![{\displaystyle \cos 3A=4\cos ^{3}A-3\cos A\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/514eab7683209efa09991eceb7496bcca761b59b)
Rumus setengah sudut
![{\displaystyle \sin {\frac {A}{2}}=\pm {\sqrt {\frac {1-\cos A}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab3d1e7d518f9b6cfd457a1c7eb3b36e0bf4a91)
![{\displaystyle \cos {\frac {A}{2}}=\pm {\sqrt {\frac {1+\cos A}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7af1f068d8bb7067451b2561ededc1b6d9bc8698)
![{\displaystyle \tan {\frac {A}{2}}=\pm {\sqrt {\frac {1-\cos A}{1+\cos A}}}={\frac {\sin A}{1+\cos A}}={\frac {1-\cos A}{\sin A}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b61a296a8b040d5814c70d32e8cbda8ee7c719d9)
Aturan Sinus, Cosinus, dan Tangen
Aturan sinus
![{\displaystyle {\frac {a}{\sin A}}\,=\,{\frac {b}{\sin B}}\,=\,{\frac {c}{\sin C}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80060897571efb3427d5f46719e7e40e46eec288)
Turunan dari aturan sinus
Luasan dari segitiga diatas dapat dirumuskan sebagai
![{\displaystyle L={\frac {1}{2}}bc\sin A={\frac {1}{2}}ac\sin B={\frac {1}{2}}ab\sin C\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/314e0a16e8e5f780d2f9fc5a63cba4f1039e4c0c)
Kalikan persamaan diatas dengan
maka akan menjadi
![{\displaystyle {\frac {2L}{abc}}={\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e77e47daf135c46e385c1af6a6d09a1f1e453353)
Aturan cosinus
![{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma \ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ecef43bc5689f468d82f86b981d291b5c48af33)
![{\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos \alpha \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/281a84fae2de83a360d7da763f8101326ea00abd)
![{\displaystyle b^{2}=a^{2}+c^{2}-2ac\cos \beta \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17241b186759a126ad848e286a325891caeb6f83)
Aturan tangen
![{\displaystyle {\frac {a-b}{a+b}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b7a4a3592aa66976f1d34bbbb403daa392f9fdf)