MathSMA

Rumus Matematika

Kumpulan rumus lengkap sesuai Kurikulum Merdeka

Kelas 10

Eksponen dan Logaritma

Sifat Eksponen

am×an=am+na^m \times a^n = a^{m+n}

Pembagian Eksponen

aman=amn\frac{a^m}{a^n} = a^{m-n}

Eksponen Negatif

an=1ana^{-n} = \frac{1}{a^n}

Akar sebagai Eksponen

an=a1n\sqrt[n]{a} = a^{\frac{1}{n}}

Sifat Logaritma

loga(b×c)=logab+logac\log_a (b \times c) = \log_a b + \log_a c

Pembagian Logaritma

logabc=logablogac\log_a \frac{b}{c} = \log_a b - \log_a c

Perpangkatan Logaritma

logabn=nlogab\log_a b^n = n \cdot \log_a b

Perubahan Basis

logab=logcblogca\log_a b = \frac{\log_c b}{\log_c a}

Barisan dan Deret

Barisan Aritmetika

Un=a+(n1)bU_n = a + (n-1)b

Deret Aritmetika

Sn=n2(a+Un)S_n = \frac{n}{2}(a + U_n)

Barisan Geometri

Un=arn1U_n = a \cdot r^{n-1}

Deret Geometri

Sn=a(rn1)r1S_n = \frac{a(r^n - 1)}{r - 1}

Deret Geometri Tak Hingga

S=a1r,r<1S_\infty = \frac{a}{1 - r}, |r| < 1

Trigonometri

Sin

sinθ=depanmiring\sin \theta = \frac{\text{depan}}{\text{miring}}

Cos

cosθ=sampingmiring\cos \theta = \frac{\text{samping}}{\text{miring}}

Tan

tanθ=depansamping\tan \theta = \frac{\text{depan}}{\text{samping}}

Identitas Pythagoras

sin2θ+cos2θ=1\sin^2 \theta + \cos^2 \theta = 1

Sudut Istimewa (0°)

sin0=0,cos0=1,tan0=0\sin 0^\circ = 0, \cos 0^\circ = 1, \tan 0^\circ = 0

Sudut Istimewa (30°)

sin30=12,cos30=32\sin 30^\circ = \frac{1}{2}, \cos 30^\circ = \frac{\sqrt{3}}{2}

Sudut Istimewa (45°)

sin45=22,cos45=22\sin 45^\circ = \frac{\sqrt{2}}{2}, \cos 45^\circ = \frac{\sqrt{2}}{2}

Sudut Istimewa (60°)

sin60=32,cos60=12\sin 60^\circ = \frac{\sqrt{3}}{2}, \cos 60^\circ = \frac{1}{2}

Fungsi Kuadrat

Bentuk Umum

f(x)=ax2+bx+c,a0f(x) = ax^2 + bx + c, a \neq 0

Diskriminan

D=b24acD = b^2 - 4ac

Akar-akar (Rumus ABC)

x1,2=b±D2ax_{1,2} = \frac{-b \pm \sqrt{D}}{2a}

Sumbu Simetri

x=b2ax = -\frac{b}{2a}

Titik Puncak

(xp,yp)=(b2a,D4a)(x_p, y_p) = \left(-\frac{b}{2a}, -\frac{D}{4a}\right)

Kelas 11

Matriks

Determinan 2×2

det(abcd)=adbc\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc

Invers 2×2

(abcd)1=1adbc(dbca)\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1} = \frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}

Perkalian Matriks

(AB)ij=k=1nAikBkj(AB)_{ij} = \sum_{k=1}^{n} A_{ik}B_{kj}

Turunan

Turunan Pangkat

ddxxn=nxn1\frac{d}{dx}x^n = nx^{n-1}

Turunan Konstanta

ddxc=0\frac{d}{dx}c = 0

Turunan Perkalian

ddx(uv)=uv+uv\frac{d}{dx}(uv) = u'v + uv'

Turunan Pembagian

ddxuv=uvuvv2\frac{d}{dx}\frac{u}{v} = \frac{u'v - uv'}{v^2}

Aturan Rantai

dydx=dydududx\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Turunan sin

ddxsinx=cosx\frac{d}{dx}\sin x = \cos x

Turunan cos

ddxcosx=sinx\frac{d}{dx}\cos x = -\sin x

Integral

Integral Tak Tentu Pangkat

xndx=xn+1n+1+C,n1\int x^n dx = \frac{x^{n+1}}{n+1} + C, n \neq -1

Integral Tentu

abf(x)dx=F(b)F(a)\int_a^b f(x) dx = F(b) - F(a)

Luas Daerah

L=ab(f(x)g(x))dxL = \int_a^b (f(x) - g(x)) dx

Volume Benda Putar

V=πab(f(x))2dxV = \pi \int_a^b (f(x))^2 dx

Lingkaran

Persamaan Lingkaran

(xa)2+(yb)2=r2(x - a)^2 + (y - b)^2 = r^2

Pusat Lingkaran

P(a,b)P(a, b)

Jari-jari

rr

Kelas 12

Statistika

Mean

xˉ=i=1nxin\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}

Median (n genap)

Me=xn2+xn2+12Me = \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2}

Modus

Mo=nilai yang paling sering munculMo = \text{nilai yang paling sering muncul}

Standar Deviasi

σ=i=1n(xixˉ)2n\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}}

Ragam (Variansi)

σ2=i=1n(xixˉ)2n\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}

Kuartil

Q1,Q2,Q3Q_1, Q_2, Q_3

Peluang

Permutasi

P(n,k)=n!(nk)!P(n, k) = \frac{n!}{(n-k)!}

Kombinasi

C(n,k)=n!k!(nk)!C(n, k) = \frac{n!}{k!(n-k)!}

Faktorial

n!=n×(n1)×(n2)××1n! = n \times (n-1) \times (n-2) \times \cdots \times 1

Peluang Kejadian

P(A)=n(A)n(S)P(A) = \frac{n(A)}{n(S)}

Peluang Komplemen

P(Ac)=1P(A)P(A^c) = 1 - P(A)

Geometri Ruang

Jarak Titik ke Titik

d=(x2x1)2+(y2y1)2+(z2z1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}

Jarak Titik ke Garis

d=ax1+by1+ca2+b2d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}

Jarak Titik ke Bidang

d=ax1+by1+cz1+da2+b2+c2d = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}

Transformasi Geometri

Translasi

A(x+a,y+b)A'(x+a, y+b)

Refleksi Sumbu X

A(x,y)A'(x, -y)

Refleksi Sumbu Y

A(x,y)A'(-x, y)

Rotasi 90°

A(y,x)A'(-y, x)

Dilatasi

A(kx,ky)A'(kx, ky)