Fungsi Trigonometri : Enam Fungsi Trigonometri Dasar dan Identitas Trigonometri
Enam Fungsi Trigonometri Dasar
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkwI4EaeOlPG1viTzS_CWu-1IL7HmrPVvDVG9UoCdSswOWe_jrOOzBZSTukglZPWkuBMRZAzClUAY9QyQFV3VpgPCx7q8Z9GwAwZ8xqkBXJ5yLiMbvv3MIAuDuumK4JZ7TD-5MKq6hbKmK/s320/bachtiar+gambar+segitiga+siku-siku.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNQbBU_ZdOOn1HiM1HsDUDtmSvXMivSQm5Ut-IIArz0V7ZXhzd1rIRgNq-TGgmH87Z_I-yQz1-7uFVBQ1mvvuKKYYtzmt58e2yNtP6f5VCvidnj1m7HyCbG4mmU8lrkprAFmNnf7H9mdbd/s1600/CodeCogsEqn+-+2020-03-25T122739.418.gif)
Rumus diatas dapat diperluas menjadi sebagai berikut:
d). Hukum Kosinus
Jawab:
1. cos (x -
) = cos x. cos
+ sin x. sin ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
= cos x . 0 + sin x. 1
= 0 + sin x
= sin x
2. sin (x +
) = sin x.cos
+ cos x. sin ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
= sin x. 0 + cos x. 1
= 0 + cos x
= cos x
3. cos (x +
) = cos x. cos
- sin x. sin ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
= cos x . 0 - sin x. 1
= 0 - sin x
= -sin x
4. sin (x -
) = sin x.cos
+ cos x. sin ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
= sin x. 0 - cos x. 1
= 0 - cos x
= -cos x
Contoh 1.2
Berikut ini adalah contoh rumus setengah sudut. Carilah nilai fungsi berikut!
Jawab:
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzPmqroVBmzsDQMB5lG7BVK1Vo7Q34vDXA7LmXUvTFubwSiZ_76zl4y9qjRoYdzYflRmfQG9iy7790JOByXTqNuOmZVePEsbpVZXq0lp7d0d-tqI36-61nmh4KsEkWfKItQ1LPQZAlnhO7/s1600/CodeCogsEqn+-+2020-03-25T133116.412.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkwI4EaeOlPG1viTzS_CWu-1IL7HmrPVvDVG9UoCdSswOWe_jrOOzBZSTukglZPWkuBMRZAzClUAY9QyQFV3VpgPCx7q8Z9GwAwZ8xqkBXJ5yLiMbvv3MIAuDuumK4JZ7TD-5MKq6hbKmK/s320/bachtiar+gambar+segitiga+siku-siku.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNQbBU_ZdOOn1HiM1HsDUDtmSvXMivSQm5Ut-IIArz0V7ZXhzd1rIRgNq-TGgmH87Z_I-yQz1-7uFVBQ1mvvuKKYYtzmt58e2yNtP6f5VCvidnj1m7HyCbG4mmU8lrkprAFmNnf7H9mdbd/s1600/CodeCogsEqn+-+2020-03-25T122739.418.gif)
Rumus diatas dapat diperluas menjadi sebagai berikut:
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcrVdrfiHe_G6DfMz1mpzZlk_fHHeSFFTlmUEc16KZBQVZSA39VjiacwZw4D-mcTy8PVmAUDSyLSJMF5e-a32-LEtoGk-mn115Agb171vBKoR60-ybMGv-IAVWPYOEMz33TnOK8eOrTJPq/s1600/CodeCogsEqn+-+2020-03-25T123542.509.gif)
Identitas Trigonometri
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhldBIAAmoNAWfD6b2H3jt9u4Lpj797uBQwRFPoDvC7wt2iJzFMelyB90A2pPSXjSv91CSM_SuUlrJPAGxHwffLIkS-O5Iw20J-RCivpBgo3BDs1ivWEoCApeyGXb93VCeR09SQ7wjK1I-r/s1600/CodeCogsEqn+-+2020-03-25T130254.727.gif)
a). Rumus Penjumlahan 2 Sudut
cos (A + B) = cos A.cos B - sin A.sin B
cos (A - B) = cos A.cos B + sin A.sin B
sin (A + B) = sin A.cos B + cos A.sin B
sin (A - B) = sin A.cos B - cos A.sin B
Dari hubungan rumus cos^2
+ sin^2
= 1 dengan rumus penjumlahan maka akan didapat rumus berikut ini.
b). Rumus Sudut-Ganda
Dan Rumus penjumlahan sendiri berasal dari penggabungan persamaan
Cos^2
+ sin^2
= 1, dan cos^2
- sin^2
= cos 2
.
Apabila kita menjumlahkan dua persamaan untuk mendapatkan 2 cos^2
= 1 + cos 2
dan mengurangkan persamaan kedua dari persamaan yang pertama untuk mendapatkan 2 sin^2
= 1 - cos 2
. Ini menghasilkan identitas berikut, yang berguna pada kalkulus integral.
c). Rumus Setengah-Sudut
d). Hukum Kosinus
Jika a, b, dan c adalah sisi segitiga ABC dan jika
adalah sudut di hadapkan c, maka:
Contoh 1.1
Buktikan!
1. cos (x -
) = sin x
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
2. sin (x +
) = cos x
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
3. cos (x +
) = -sin x
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
4. sin (x -
) = -cos x
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
Jawab:
1. cos (x -
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
= cos x . 0 + sin x. 1
= 0 + sin x
= sin x
2. sin (x +
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
= sin x. 0 + cos x. 1
= 0 + cos x
= cos x
3. cos (x +
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
= cos x . 0 - sin x. 1
= 0 - sin x
= -sin x
4. sin (x -
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJNuEbjMshxJKzAxtpzZjuujqgqOVr-1Sbg7Ffgkv7-EtKsriuuU0AXk6H5ZE9j8V5jfP-Lgo2Xs0ywbMvwYg3_o6ex36Si_x9YuWww5RAoef3rLhnb7zTOIA-QOibjRpUgfqu0A8aiPSw/s1600/CodeCogsEqn+-+2020-03-25T131442.226.gif)
= sin x. 0 - cos x. 1
= 0 - cos x
= -cos x
Contoh 1.2
Berikut ini adalah contoh rumus setengah sudut. Carilah nilai fungsi berikut!
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhY1CnHYtlDudLeBYTc2KvCNy6nLerSFIzmVR7vbBdROzm_3N34uKOdyFR1T-DgBEEXZlTTvP6gE58lkw-k2eU6En7nkf82c8bWxmiiweJnVCVs10mO6yBFg3EAeWSXAOdqaseqIKfevAtc/s1600/CodeCogsEqn+-+2020-03-25T131939.608.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhP6upnq_VjrjKoJqUsja2fg7y88PPWvtnSdU3e0c-xKxccaQEYvLhLESpt5uwzFp15kavrdMJsqs3iU8SP02kZYRNVuE-266sKLywek00XVxJYXBBvcH6jv6D8HjWzJ5-NzemBSWg_AeIP/s1600/CodeCogsEqn+-+2020-03-25T132256.504.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcK5_zDIkIqwwx5fb2YwqJIxQ90kwa7At4UVXoVoZ2Gj7izT_aXT6aJa8xVlW7fh7bwb3cdf6nJOxw1CzVUW-0TfrvXx1MIKIgzobNpRbVzpjghWNzI9srNrTmgemDOgbRa5x8yBL-Lhh2/s1600/CodeCogsEqn+-+2020-03-25T132503.541.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6TFcPJst1zuYmQMnLz_IS9qUbcN_L5h3ecVhyPlRKr1b8xi0u-hZyP9O5FSMni-8l6p4IneaUL7U-r59OOnVtrGq2yvX6YJODfjdxBqRTxXyQnPve1O2BtnO_BlaxJBsMT-6s2UkkN7TL/s1600/CodeCogsEqn+-+2020-03-25T132756.612.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzPmqroVBmzsDQMB5lG7BVK1Vo7Q34vDXA7LmXUvTFubwSiZ_76zl4y9qjRoYdzYflRmfQG9iy7790JOByXTqNuOmZVePEsbpVZXq0lp7d0d-tqI36-61nmh4KsEkWfKItQ1LPQZAlnhO7/s1600/CodeCogsEqn+-+2020-03-25T133116.412.gif)