Log InSign Up

HSC

Mathematics Extension 2Formula Sheet

Measurement

Length

l=θ360×2πr\displaystyle l = \frac{\theta}{360} \times 2\pi r

Area

A=θ360×πr2\displaystyle A = \frac{\theta}{360} \times \pi r^{2}
A=h2 (a+b)\displaystyle A = \frac{h}{2}\,(a + b)

Surface area

A=2πr2+2πrh\displaystyle A = 2\pi r^{2} + 2\pi r h
A=4πr2\displaystyle A = 4\pi r^{2}

Volume

V=13Ah\displaystyle V = \frac{1}{3} A h
V=43πr3\displaystyle V = \frac{4}{3} \pi r^{3}
Functions
x=−b±b2−4ac2a\displaystyle x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
For ax3+bx2+cx+d=0:α+β+γ=−baαβ+αγ+βγ=caandαβγ=−da\displaystyle \begin{aligned} \text{For } ax^{3} + bx^{2} + cx + d & = 0: \\ \alpha + \beta + \gamma & = -\frac{b}{a} \\ \alpha\beta + \alpha\gamma + \beta\gamma & = \frac{c}{a} \\ \text{and} \hspace{0.5em} \alpha\beta\gamma & = -\frac{d}{a} \end{aligned}

Relations

(x−h)2+(y−k)2=r2\displaystyle (x - h)^{2} + (y - k)^{2} = r^{2}
Financial Mathematics
A=P(1+r)n\displaystyle A = P(1 + r)^{n}

Sequences and series

Tn=a+(n−1)d\displaystyle T_{n} = a + (n - 1)d
Sn=n2[2a+(n−1)d]=n2(a+l)\displaystyle S_{n} = \frac{n}{2}\left[2a + (n - 1)d\right] = \frac{n}{2}(a + l)
Tn=a rn−1\displaystyle T_{n} = a\,r^{n-1}
Sn=a(1−rn)1−r=a(rn−1)r−1,  r≠1\displaystyle S_{n} = \frac{a(1 - r^{n})}{1 - r} = \frac{a(r^{n} - 1)}{r - 1},\ \ r \ne 1
S=a1−r,  ∣r∣<1\displaystyle S = \frac{a}{1 - r},\ \ |r| < 1
Logarithmic and Exponential Functions
log⁡aax=x=alog⁡ax\displaystyle \log_{a} a^{x} = x = a^{\log_{a} x}
log⁡ax=log⁡bxlog⁡ba\displaystyle \log_{a} x = \frac{\log_{b} x}{\log_{b} a}
ax=exln⁡a\displaystyle a^{x} = e^{x\ln a}
Trigonometric Functions
sin⁡A=opphyp,  cos⁡A=adjhyp,  tan⁡A=oppadj\displaystyle \sin A = \frac{\mathrm{opp}}{\mathrm{hyp}},\; \cos A = \frac{\mathrm{adj}}{\mathrm{hyp}},\; \tan A = \frac{\mathrm{opp}}{\mathrm{adj}}
A=12absin⁡C\displaystyle A = \frac{1}{2} ab \sin C
asin⁡A=bsin⁡B=csin⁡C\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
c2=a2+b2−2abcos⁡C\displaystyle c^{2} = a^{2} + b^{2} - 2ab \cos C
cos⁡C=a2+b2−c22ab\displaystyle \cos C = \frac{a^{2} + b^{2} - c^{2}}{2ab}
l=rθ\displaystyle l = r \theta
A=12r2θ\displaystyle A = \frac{1}{2}r^2\theta
Isoceles triangle
Special right-angled triangle

Trigonometric identities

sec⁡A=1cos⁡A,  cos⁡A≠0\displaystyle \sec A = \frac{1}{\cos A},\; \cos A \ne 0
cosec⁡A=1sin⁡A,  sin⁡A≠0\displaystyle \cosec A = \frac{1}{\sin A},\; \sin A \ne 0
cot⁡A=cos⁡Asin⁡A,  sin⁡A≠0\displaystyle \cot A = \frac{\cos A}{\sin A},\; \sin A \ne 0
cos⁡2x+sin⁡2x=1\displaystyle \cos^{2} x + \sin^{2} x = 1

Compound angles

sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\displaystyle \sin(A + B) = \sin A \cos B + \cos A \sin B
cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\displaystyle \cos(A + B) = \cos A \cos B - \sin A \sin B
tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\displaystyle \tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}
If t=tan⁡A2 then: sin⁡A=2t1+t2cos⁡A=1−t21+t2tan⁡A=2t1−t2\displaystyle \text{If } t = \tan\frac{A}{2} \text{ then: } \\ \begin{aligned} \hphantom{\hspace{2em}}\sin A & = \frac{2t}{1 + t^{2}} \\ \hphantom{\hspace{2em}}\cos A & = \frac{1 - t^{2}}{1 + t^{2}} \\ \hphantom{\hspace{2em}}\tan A & = \frac{2t}{1 - t^{2}} \end{aligned}
cos⁡Acos⁡B=12[cos⁡(A−B)+cos⁡(A+B)]\displaystyle \cos A \cos B = \frac{1}{2}\bigl[\cos(A - B) + \cos(A + B)\bigr]
sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)]\displaystyle \sin A \sin B = \frac{1}{2}\bigl[\cos(A - B) - \cos(A + B)\bigr]
sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\displaystyle \sin A \cos B = \frac{1}{2}\bigl[\sin(A + B) + \sin(A - B)\bigr]
cos⁡Asin⁡B=12[sin⁡(A+B)−sin⁡(A−B)]\displaystyle \cos A \sin B = \frac{1}{2}\bigl[\sin(A + B) - \sin(A - B)\bigr]
sin⁡2nx=12(1−cos⁡2nx)\displaystyle \sin^{2}nx = \frac{1}{2}\bigl(1 - \cos2nx\bigr)
cos⁡2nx=12(1+cos⁡2nx)\displaystyle \cos^{2}nx = \frac{1}{2}\bigl(1 + \cos2nx\bigr)
Statistical Analysis
z=x−μσ\displaystyle z = \frac{x - \mu}{\sigma}
An outlier is a scoreless than Q1−1.5×IQRor more than Q3+1.5×IQR\displaystyle \begin{aligned} \text{An outlier is a score} \\ \text{less than } & Q_{1} - 1.5 \times IQR \\ \text{or more than } & Q_{3} + 1.5 \times IQR \end{aligned}

Normal distribution

Normal distribution curve
Approximately 68% of scores have z-scores between -1 and 1
Approximately 95% of scores have z-scores between -2 and 2
Approximately 99.7% of scores have z-scores between -3 and 3
E(X)=μ\displaystyle E(X) = \mu
Var⁡(X)=E[(X−μ)2]=E(X2)−μ2\displaystyle \operatorname{Var}(X) = E\big[(X - \mu)^2\big] = E(X^{2}) - \mu^{2}

Probability

P(A∩B)=P(A)P(B)\displaystyle P(A \cap B) = P(A)P(B)
P(A∪B)=P(A)+P(B)−P(A∩B)\displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B)
P(A∣B)=P(A∩B)P(B),  P(B)≠0\displaystyle P(A\mid B) = \frac{P(A \cap B)}{P(B)},\; P(B) \ne 0

Continuous random variables

P(X≤r)=∫arf(x) dx\displaystyle P(X \le r) = \int_{a}^{r} f(x)\,dx
P(a<X<b)=∫abf(x) dx\displaystyle P(a < X < b) = \int_{a}^{b} f(x)\,dx

Binomial distribution

P(X=r)=nCr pr(1−p)n−r\displaystyle P(X = r) = {}^{n}C_{r}\, p^{r}(1 - p)^{n-r}
X∼Bin⁡(n,p)\displaystyle X \sim \operatorname{Bin}(n, p)
⇒  P(X=x)=(nx)px(1−p)n−x,  x=0,1,…,n\displaystyle \Rightarrow\; P(X = x) = \binom{n}{x} p^{x}(1 - p)^{n-x},\; x = 0, 1, \ldots, n
E(X)=np\displaystyle E(X) = np
Var⁡(X)=np(1−p)\displaystyle \operatorname{Var}(X) = np(1 - p)
Differential Calculus
FunctionDerivative
y=f(x)n\displaystyle y = f(x)^{n}dydx=nf′(x)[f(x)]n−1\displaystyle \frac{dy}{dx} = n f'(x)[f(x)]^{n-1}
y=uv\displaystyle y = uvdydx=u dvdx+v dudx\displaystyle \frac{dy}{dx} = u\,\frac{dv}{dx} + v\,\frac{du}{dx}
y=g(u)  where  u=f(x)\displaystyle y = g(u)\,\text{ where }\, u = f(x)dydx=dydu×dudx\displaystyle \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}
y=uv\displaystyle y = \dfrac{u}{v}dydx=v dudx−u dvdxv2\displaystyle \frac{dy}{dx} = \dfrac{v\,\dfrac{du}{dx} - u\,\dfrac{dv}{dx}}{v^{2}}
y=sin⁡f(x)\displaystyle y = \sin f(x)dydx=f′(x)cos⁡f(x)\displaystyle \frac{dy}{dx} = f'(x)\cos f(x)
y=cos⁡f(x)\displaystyle y = \cos f(x)dydx=− f′(x)sin⁡f(x)\displaystyle \frac{dy}{dx} = -\,f'(x)\sin f(x)
y=tan⁡f(x)\displaystyle y = \tan f(x)dydx=f′(x)sec⁡2f(x)\displaystyle \frac{dy}{dx} = f'(x)\sec^{2} f(x)
y=ef(x)\displaystyle y = e^{f(x)}dydx=f′(x)ef(x)\displaystyle \frac{dy}{dx} = f'(x)e^{f(x)}
y=ln⁡f(x)\displaystyle y = \ln f(x)dydx=f′(x)f(x)\displaystyle \frac{dy}{dx} = \dfrac{f'(x)}{f(x)}
y=af(x)\displaystyle y = a^{f(x)}dydx=(ln⁡a) f′(x) af(x)\displaystyle \frac{dy}{dx} = (\ln a)\,f'(x)\,a^{f(x)}
y=log⁡af(x)\displaystyle y = \log_{a} f(x)dydx=f′(x)(ln⁡a) f(x)\displaystyle \frac{dy}{dx} = \dfrac{f'(x)}{(\ln a)\,f(x)}
y=sin⁡−1f(x)\displaystyle y = \sin^{-1} f(x)dydx=f′(x)1−[f(x)]2\displaystyle \frac{dy}{dx} = \dfrac{f'(x)}{\sqrt{1 - [f(x)]^{2}}}
y=cos⁡−1f(x)\displaystyle y = \cos^{-1} f(x)dydx=− f′(x)1−[f(x)]2\displaystyle \frac{dy}{dx} = -\,\dfrac{f'(x)}{\sqrt{1 - [f(x)]^{2}}}
y=tan⁡−1f(x)\displaystyle y = \tan^{-1} f(x)dydx=f′(x)1+[f(x)]2\displaystyle \frac{dy}{dx} = \dfrac{f'(x)}{1 + [f(x)]^{2}}
Integral Calculus
∫f′(x)[f(x)]n dx=1n+1[f(x)]n+1+cwhere n≠−1\displaystyle \int f'(x)[f(x)]^{n}\,dx = \frac{1}{n+1}[f(x)]^{n+1} + c \\ \text{where } n \ne -1
∫f′(x)sin⁡f(x) dx=−cos⁡f(x)+c\displaystyle \int f'(x)\sin f(x)\,dx = -\cos f(x) + c
∫f′(x)cos⁡f(x) dx=sin⁡f(x)+c\displaystyle \int f'(x)\cos f(x)\,dx = \sin f(x) + c
∫f′(x)sec⁡2f(x) dx=tan⁡f(x)+c\displaystyle \int f'(x)\sec^{2} f(x)\,dx = \tan f(x) + c
∫f′(x)ef(x) dx=ef(x)+c\displaystyle \int f'(x)e^{f(x)}\,dx = e^{f(x)} + c
∫f′(x)f(x) dx=ln⁡∣f(x)∣+c\displaystyle \int \frac{f'(x)}{f(x)}\,dx = \ln\lvert f(x)\rvert + c
∫f′(x) af(x) dx=af(x)ln⁡a+c\displaystyle \int f'(x)\,a^{f(x)}\,dx = \frac{a^{f(x)}}{\ln a} + c
∫f′(x)a2−[f(x)]2 dx=sin⁡−1f(x)a+c\displaystyle \int \frac{f'(x)}{\sqrt{a^{2} - [f(x)]^{2}}}\,dx = \sin^{-1} \frac{f(x)}{a} + c
∫f′(x)a2+[f(x)]2 dx=1atan⁡−1f(x)a+c\displaystyle \int \frac{f'(x)}{a^{2} + [f(x)]^{2}}\,dx = \frac{1}{a}\tan^{-1} \frac{f(x)}{a} + c
∫u dvdx dx=uv−∫v dudx dx\displaystyle \int u\,\frac{dv}{dx}\,dx = u v - \int v\,\frac{du}{dx}\,dx
∫abf(x) dx≈ ⁣b−a2n ⁣{f(a)+f(b) +2[f(x1) ⁣+ ⁣⋯ ⁣+ ⁣f(xn−1)] ⁣}where a=x0 and b=xn\displaystyle \begin{aligned} & \int_{a}^{b} f(x)\,dx \\[6pt] & \approx \! \frac{b-a}{2n} \! \Bigg\{\begin{aligned} & f(a) + f(b) \, + \\[3pt] & 2\big[ f(x_{1}) \! + \! \dotsb \! + \! f(x_{n-1}) \big] \! \end{aligned}\Bigg\} \\[6pt] & \text{where } a = x_{0} \text{ and } b = x_{n} \end{aligned}
Combinatorics
nPr=n!(n−r)!\displaystyle {}^{n}P_{r} = \dfrac{n!}{(n-r)!}
(nr)=nCr=n!r!(n−r)!\displaystyle \binom{n}{r} = {}^{n}C_{r} = \dfrac{n!}{r!(n-r)!}
(x+a)n=xn+(n1)xn−1a+⋯+(nr)xn−rar+⋯+an\displaystyle (x + a)^{n} = x^{n} + \binom{n}{1}x^{n-1}a + \cdots + \binom{n}{r}x^{n-r}a^{r} + \cdots + a^{n}
Vectors
∣u∼∣=∣x i∼+y j∼∣=x2+y2\displaystyle \lvert\underset{\sim}{u}\rvert = \lvert x\,\underset{\sim}{i} + y\,\underset{\sim}{j}\rvert = \sqrt{x^{2} + y^{2}}
u∼ ⁣⋅ ⁣v∼=∣u∼∣ ∣v∼∣cos⁡θ=x1x2+y1y2,where u∼=x1 i∼+y1 j∼and v∼=x2 i∼+y2 j∼\displaystyle \underset{\sim}{u}\!\cdot\!\underset{\sim}{v} = \lvert\underset{\sim}{u}\rvert\,\lvert\underset{\sim}{v}\rvert\cos\theta = x_{1}x_{2} + y_{1}y_{2}, \\\begin{aligned} \text{where }\underset{\sim}{u}=x_{1}\,\underset{\sim}{i}+y_{1}\,\underset{\sim}{j}\\\text{and }\underset{\sim}{v}=x_{2}\,\underset{\sim}{i}+y_{2}\,\underset{\sim}{j}\end{aligned}
r=a+λb\displaystyle r = a + \lambda b
Complex Numbers
z=a+ib=r(cos⁡θ+isin⁡θ)=reiθ\displaystyle z = a + ib = r(\cos\theta + i\sin\theta) = r e^{i\theta}
[r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)=rneinθ\displaystyle \big[r(\cos\theta + i\sin\theta)\big]^{n} = r^{n}(\cos n\theta + i\sin n\theta) = r^{n}e^{in\theta}
Mechanics
d2xdt2=dvdt=v dvdx=ddx ⁣(12v2)\displaystyle \frac{d^{2}x}{dt^{2}} = \frac{dv}{dt} = v\,\frac{dv}{dx} = \frac{d}{dx}\!\left(\frac{1}{2}v^{2}\right)
x=acos⁡(nt+α)+c\displaystyle x = a\cos(nt + \alpha) + c
x=asin⁡(nt+α)+c\displaystyle x = a\sin(nt + \alpha) + c
x¨=−n2(x−c)\displaystyle \ddot{x} = -n^{2}(x - c)