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 kekerabatan kuadran I (tiruana +)
sin(90° − α) = cos α
cos(90° − α) = sin α
tan(90° − α) = cot α

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

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

Sudut kekerabatan 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 = \(\frac{b}{a}\)