Rumus-Rumus Trigonometri

Perbandingan Trigonometri pada Segitiga Siku-Siku

sin α = $\mathrm{\frac{y}{r}}$       csc α = $\mathrm{\frac{r}{y}}$
cos α = $\mathrm{\frac{x}{r}}$       sec α = $\mathrm{\frac{r}{x}}$
tan α = $\mathrm{\frac{y}{x}}$       cot α = $\mathrm{\frac{x}{y}}$


Identitas Trigonometri

sin α = $\mathrm{\frac{1}{csc\,\alpha}}$   ⇔   csc α = $\mathrm{\frac{1}{sin\,\alpha}}$
cos α = $\mathrm{\frac{1}{sec\,\alpha}}$   ⇔   sec α = $\mathrm{\frac{1}{cos\,\alpha}}$
tan α = $\mathrm{\frac{1}{cot\,\alpha}}$   ⇔   cot α = $\mathrm{\frac{1}{tan\,\alpha}}$

tan α = $\mathrm{\frac{sin\,\alpha}{cos\,\alpha}}$
cot α = $\mathrm{\frac{cos\,\alpha}{sin\,\alpha}}$

sin²α + cos²α = 1
tan²α + 1 = sec²α
1 + cot²α = csc²α

Perbandingan Trigonometri Sudut Berelasi
Sudut relasi kuadran I (semua +)
sin(90° − α) = cos α
cos(90° − α) = sin α
tan(90° − α) = cot α

Sudut relasi kuadran II (sin +)
sin(180° − α) = sin α
cos(180° − α) = -cos α
tan(180° − α) = -tan α

Sudut relasi kuadran III (tan +)
sin(180° + α) = -sin α
cos(180° + α) = -cos α
tan(180° + α) = tan α

Sudut relasi Kuadran IV (cos +)
sin(360° − α) = -sin α
cos(360° − α) = cos α
tan(360° − α) = -tan α

Sudut Negatif :
sin(-α) = -sin α
cos(-α) = cos α
tan(-α) = -tan α

Sudut > 360° :
sin (α + n.360°) = sin α
cos (α + n.360°) = cos α
tan (α + n.360°) = tan α
n bilangan bulat


Aturan Sinus dan Cosinus

Aturan sinus :
$\mathrm{\frac{\mathit{a}}{sin\,A}=\frac{\mathit{b}}{sin\,B}=\frac{\mathit{c}}{sin\,C}}$

Aturan Cosinus :
a² = b² + c² − 2bc cos A
b² = a² + c² − 2ac cos B
c² = a² + b² − 2ab cos C


Rumus Luas Segitiga

L = $\frac{1}{2}$bc sin A
L = $\frac{1}{2}$ac sin B
L = $\frac{1}{2}$ab sin C

L = $\mathrm{\frac{\mathit{a}^{2}\,sin\,B\,sin\,C}{2\,sin\,A}}$
L = $\mathrm{\frac{\mathit{b}^{2}\,sin\,A\,sin\,C}{2\,sin\,B}}$
L = $\mathrm{\frac{\mathit{c}^{2}\,sin\,A\,sin\,B}{2\,sin\,C}}$

Rumus Jumlah dan Selisih Dua Sudut
sin(α + β) = sin α cos β + cos α sin β
sin(α − β) = sin α cos β − cos α sin β

cos(α + β) = cos α cos β − sin α sin β
cos(α − β) = cos α cos β + sin α sin β

tan(α + β) = $\mathrm{\frac{tan\,\alpha \,+\,tan\,\beta }{1\,-\,tan\,\alpha \,tan\,\beta }}$
tan(α − β) = $\mathrm{\frac{tan\,\alpha \,-\,tan\,\beta }{1\,+\,tan\,\alpha \,tan\,\beta }}$


Rumus Sudut Rangkap Dua
sin 2α = 2 sin α cos α
cos 2α = cos²α − sin²α
cos 2α = 2cos²α − 1
cos 2α = 1 − 2sin²α
tan 2α = $\mathrm{\frac{2\,tan\,\alpha }{1-tan^{2}\alpha }}$


Rumus Setengah Sudut
sin$\frac{1}{2}$α = $\mathrm{\pm \sqrt{\frac{1-cos\,\alpha }{2}}}$
cos$\frac{1}{2}$α = $\mathrm{\pm \sqrt{\frac{1+cos\,\alpha }{2}}}$
tan$\frac{1}{2}$α = $\mathrm{\pm \sqrt{\frac{1-cos\,\alpha }{1+cos\,\alpha }}}$,  cos α ≠ -1
tan$\frac{1}{2}$α = $\mathrm{\frac{1-cos\,\alpha }{sin\,\alpha }}$,  sin α ≠ 0
tan$\frac{1}{2}$α = $\mathrm{\frac{sin\,\alpha }{1+cos\,\alpha }}$,  cos α ≠ -1


Rumus Perkalian Sinus dan Cosinus
2sin α cos β = sin(α + β) + sin(α − β)
2cos α sin β = sin(α + β) − sin(α − β)
2cos α cos β = cos(α + β) + cos(α − β)
-2sin α sin β = cos(α + β) − cos(α − β)


Rumus Jumlah dan Selisih Sinus dan Cosinus
sin α + sin β = 2sin$\left ( \frac{\alpha +\beta }{2} \right )$ cos$\left ( \frac{\alpha -\beta }{2} \right )$
sin α − sin β = 2cos$\left ( \frac{\alpha +\beta }{2} \right )$ sin$\left ( \frac{\alpha -\beta }{2} \right )$
cos α + cos β = 2cos$\left ( \frac{\alpha +\beta }{2} \right )$ cos$\left ( \frac{\alpha -\beta }{2} \right )$
cos α − cos β = -2sin$\left ( \frac{\alpha +\beta }{2} \right )$ sin$\left ( \frac{\alpha -\beta }{2} \right )$


Persamaan Trigonometri
sin x = sin α
     x = α + k.360° atau
     x = (180° − α) + k.360°

cos x = cos α
     x = α + k.360° atau
     x = -α + k.360°

tan x = tan α
     x = α + k.180°
     dengan k bilangan bulat


Bentuk acos x + bsin x
acos x + bsin x = k cos(x − θ)
dengan :
k = $\sqrt{a^{2}+b^{2}}$
tan θ = $\frac{b}{a}$