Materi: Matematika (Trigonometri)
Tabel Trigonometri Sudut-Sudut Istimewa
| Fungsi | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | \( \frac{1}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{\sqrt{3}}{2} \) | 1 |
| cos | 1 | \( \frac{\sqrt{3}}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{1}{2} \) | 0 |
| tan | 0 | \( \frac{\sqrt{3}}{3} \) | 1 | \( \sqrt{3} \) | – |
| cosec | – | 2 | \( \sqrt{2} \) | \( \frac{2}{\sqrt{3}} \) | 1 |
| sec | 1 | \( \frac{2}{\sqrt{3}} \) | \( \sqrt{2} \) | 2 | – |
| cot | – | \( \sqrt{3} \) | 1 | \( \frac{\sqrt{3}}{3} \) | 0 |
| \( \theta \) (°) | \( \sin\theta \) (desimal) | \( \sin\theta \) (eksak) | \( \cos\theta \) (desimal) | \( \cos\theta \) (eksak) | \( \tan\theta \) (desimal) | \( \tan\theta \) (eksak) |
|---|---|---|---|---|---|---|
| 0 | 0.000000 | \(0\) | 1.000000 | \(1\) | 0.000000 | \(0\) |
| 15 | 0.258819 | \(\tfrac{\sqrt6-\sqrt2}{4}\) | 0.965926 | \(\tfrac{\sqrt6+\sqrt2}{4}\) | 0.267949 | \(2-\sqrt3\) |
| 26.565 | 0.447214 | \(\tfrac{1}{\sqrt5}\) | 0.894427 | \(\tfrac{2}{\sqrt5}\) | 0.500000 | \(\tfrac{1}{2}\) |
| 30 | 0.500000 | \(\tfrac{1}{2}\) | 0.866025 | \(\tfrac{\sqrt3}{2}\) | 0.577350 | \(\tfrac{1}{\sqrt3}\) |
| 36.870 | 0.600000 | \(\tfrac{3}{5}\) | 0.800000 | \(\tfrac{4}{5}\) | 0.750000 | \(\tfrac{3}{4}\) |
| 45 | 0.707106 | \(\tfrac{\sqrt2}{2}\) | 0.707106 | \(\tfrac{\sqrt2}{2}\) | 1.000000 | \(1\) |
| 53.130 | 0.800000 | \(\tfrac{4}{5}\) | 0.600000 | \(\tfrac{3}{5}\) | 1.333333 | \(\tfrac{4}{3}\) |
| 60 | 0.866025 | \(\tfrac{\sqrt3}{2}\) | 0.500000 | \(\tfrac{1}{2}\) | 1.732051 | \(\sqrt3\) |
| 90 | 1.000000 | \(1\) | 0.000000 | \(0\) | — | — |
A. Rumus Perbandingan Trigonometri
Rumus perbandingan trigonometri adalah sebagai berikut:
\[ \sin \alpha = \frac{\text{sisi depan}}{\text{sisi miring}} = \frac{y}{r} \] \[ \cos \alpha = \frac{\text{sisi samping}}{\text{sisi miring}} = \frac{x}{r} \] \[ \tan \alpha = \frac{\text{sisi depan}}{\text{sisi samping}} = \frac{y}{x} \]
- \(\csc \alpha = \frac{1}{\sin \alpha} = \frac{r}{y}\)
- \(\sec \alpha = \frac{1}{\cos \alpha} = \frac{r}{x}\)
- \(\cot \alpha = \frac{1}{\tan \alpha} = \frac{x}{y}\)
dan berlaku teorema Phytagoras, yaitu:
\[ x^2 + y^2 = r^2 \]
Dengan menggunakan hubungan Phytagoras:
\[ \sin^2 \alpha + \cos^2 \alpha = 1 \] \[ \tan^2 \alpha + 1 = \sec^2 \alpha \] \[ 1 + \cot^2 \alpha = \csc^2 \alpha \]
B. Nilai-nilai Sudut Istimewa
Nilai fungsi trigonometri untuk sudut-sudut istimewa adalah sebagai berikut:
| 0° | 30° | 45° | 60° | 90° | |
|---|---|---|---|---|---|
| sin | 0 | \(\frac{1}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{\sqrt{3}}{2}\) | 1 |
| cos | 1 | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{2}\) | 0 |
| tan | 0 | \(\frac{1}{\sqrt{3}}\) | 1 | \(\sqrt{3}\) | \(\infty\) |
KUADRAN
Untuk menentukan nilai fungsi trigonometri sudut istimewa yang lebih dari 90° dapat digunakan rumusan relasi kuadran di bawah ini:
Sudut = (\(\alpha \pm k \cdot 90\)°)
Dengan ketentuan:
- k genap maka fungsi tetap:
sin ⇒ sin
cos ⇒ cos
tan ⇒ tan - k ganjil maka fungsi berubah:
sin ⇒ cos
cos ⇒ sin
tan ⇒ cotan
Tanda negatif dan positif tergantung kuadran fungsi asal.
C. Dalil-dalil dalam Segitiga
Perhatikan gambar berikut!
- Dalil Sinus
\[ \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \]
- Dalil Kosinus
\[ a^2 = b^2 + c^2 - 2bc \cos \alpha \] \[ b^2 = a^2 + c^2 - 2ac \cos \beta \] \[ c^2 = a^2 + b^2 - 2ab \cos \gamma \]
- Luas Segitiga ABC
\[ L = \frac{1}{2} b \cdot c \cdot \sin \alpha = \frac{1}{2} a \cdot c \cdot \sin \beta = \frac{1}{2} a \cdot b \cdot \sin \gamma \]
D. Rumus Penjumlahan Sudut
\[ \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \] \[ \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \] \[ \tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \cdot \tan b} \]
\[ \sin 2a = 2 \sin a \cos a \] \[ \cos 2a = \cos^2 a - \sin^2 a \] \[ \cos 2a = 2 \cos^2 a - 1 \] \[ \cos 2a = 1 - 2 \sin^2 a \] \[ \tan 2a = \frac{2 \tan a}{1 - \tan^2 a} \] \[ \sin^2 a + \cos^2 a = 1 \] \[ 1 + \tan^2 a = \sec^2 a \]
E. Rumus Jumlah, Selisih, Kali, dan Bagi
- \(\sin A + \sin B = 2 \sin \frac{1}{2}(A + B) \cos \frac{1}{2}(A - B)\)
- \(\sin A - \sin B = 2 \cos \frac{1}{2}(A + B) \sin \frac{1}{2}(A - B)\)
- \(\cos A + \cos B = 2 \cos \frac{1}{2}(A + B) \cos \frac{1}{2}(A - B)\)
- \(\cos A - \cos B = -2 \sin \frac{1}{2}(A + B) \sin \frac{1}{2}(A - B)\)
- \(2 \sin A \cos B = \sin(A + B) + \sin(A - B)\)
- \(2 \cos A \sin B = \sin(A + B) - \sin(A - B)\)
- \(2 \cos A \cos B = \cos(A + B) + \cos(A - B)\)
- \(-2 \sin A \sin B = \cos(A + B) - \cos(A - B)\)
| Sudut | Radian | \(\sin\) | \(\cos\) | \(\tan\) |
|---|---|---|---|---|
| \(-90^\circ\) | \(-\tfrac{\pi}{2}\) | \(-1\) | \(0\) | tak terdefinisi |
| \(-60^\circ\) | \(-\tfrac{\pi}{3}\) | \(-\tfrac{\sqrt{3}}{2}\) | \(\tfrac{1}{2}\) | \(-\sqrt{3}\) |
| \(-45^\circ\) | \(-\tfrac{\pi}{4}\) | \(-\tfrac{\sqrt{2}}{2}\) | \(\tfrac{\sqrt{2}}{2}\) | \(-1\) |
| \(-30^\circ\) | \(-\tfrac{\pi}{6}\) | \(-\tfrac{1}{2}\) | \(\tfrac{\sqrt{3}}{2}\) | \(-\tfrac{\sqrt{3}}{3}\) |
| \(0^\circ\) | \(0\) | \(0\) | \(1\) | \(0\) |
| \(30^\circ\) | \(\tfrac{\pi}{6}\) | \(\tfrac{1}{2}\) | \(\tfrac{\sqrt{3}}{2}\) | \(\tfrac{\sqrt{3}}{3}\) |
| \(45^\circ\) | \(\tfrac{\pi}{4}\) | \(\tfrac{\sqrt{2}}{2}\) | \(\tfrac{\sqrt{2}}{2}\) | \(1\) |
| \(60^\circ\) | \(\tfrac{\pi}{3}\) | \(\tfrac{\sqrt{3}}{2}\) | \(\tfrac{1}{2}\) | \(\sqrt{3}\) |
| \(90^\circ\) | \(\tfrac{\pi}{2}\) | \(1\) | \(0\) | tak terdefinisi |
| \(y\) | \(\arcsin(y)\) rad | \(\arcsin(y)\) derajat | \(\arccos(y)\) rad | \(\arccos(y)\) derajat |
|---|---|---|---|---|
| \(-1\) | \(-\tfrac{\pi}{2}\) | \(-90^\circ\) | \(\pi\) | \(180^\circ\) |
| \(-\tfrac{\sqrt{3}}{2}\) | \(-\tfrac{\pi}{3}\) | \(-60^\circ\) | \(\tfrac{5\pi}{6}\) | \(150^\circ\) |
| \(-\tfrac{\sqrt{2}}{2}\) | \(-\tfrac{\pi}{4}\) | \(-45^\circ\) | \(\tfrac{3\pi}{4}\) | \(135^\circ\) |
| \(-\tfrac{1}{2}\) | \(-\tfrac{\pi}{6}\) | \(-30^\circ\) | \(\tfrac{2\pi}{3}\) | \(120^\circ\) |
| \(0\) | \(0\) | \(0^\circ\) | \(\tfrac{\pi}{2}\) | \(90^\circ\) |
| \(\tfrac{1}{2}\) | \(\tfrac{\pi}{6}\) | \(30^\circ\) | \(\tfrac{\pi}{3}\) | \(60^\circ\) |
| \(\tfrac{\sqrt{2}}{2}\) | \(\tfrac{\pi}{4}\) | \(45^\circ\) | \(\tfrac{\pi}{4}\) | \(45^\circ\) |
| \(\tfrac{\sqrt{3}}{2}\) | \(\tfrac{\pi}{3}\) | \(60^\circ\) | \(\tfrac{\pi}{6}\) | \(30^\circ\) |
| \(1\) | \(\tfrac{\pi}{2}\) | \(90^\circ\) | \(0\) | \(0^\circ\) |
| \(y\) | \(\arctan(y)\) rad | \(\arctan(y)\) derajat |
|---|---|---|
| \(-\sqrt{3}\) | \(-\tfrac{\pi}{3}\) | \(-60^\circ\) |
| \(-1\) | \(-\tfrac{\pi}{4}\) | \(-45^\circ\) |
| \(-\tfrac{\sqrt{3}}{3}\) | \(-\tfrac{\pi}{6}\) | \(-30^\circ\) |
| \(0\) | \(0\) | \(0^\circ\) |
| \(\tfrac{\sqrt{3}}{3}\) | \(\tfrac{\pi}{6}\) | \(30^\circ\) |
| \(1\) | \(\tfrac{\pi}{4}\) | \(45^\circ\) |
| \(\sqrt{3}\) | \(\tfrac{\pi}{3}\) | \(60^\circ\) |
Rumus Setengah Sudut
\(\displaystyle \sin\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{2}}\)
\(\displaystyle \cos\frac{\theta}{2}=\pm\sqrt{\frac{1+\cos\theta}{2}}\)
\(\displaystyle \tan\frac{\theta}{2} =\pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}} =\frac{1-\cos\theta}{\sin\theta} =\frac{\sin\theta}{1+\cos\theta}\)
Catatan tanda (\(\pm\)): pilih tanda sesuai kuadran \(\tfrac{\theta}{2}\). Untuk \(\;0\le\theta\le\pi\Rightarrow 0\le\tfrac{\theta}{2}\le\tfrac{\pi}{2}\): semua bernilai positif.