Formulas not on formula booklet This is a list of Analysis and Approaches (AA) formulas and properties that IB expects you to know, but are not listed in the formula booklet. These formulas do not need to be derived on exams.
Published on 2026-08-20
Contents Prior Learning and Algebra Assume x , y > 0 x, y > 0 x , y > 0
formula / property description a 0 = 1 a^0 = 1 a 0 = 1 a ≠ 0 a \neq 0 a = 0 x a ⋅ x b = x a + b x^a \cdot x^b = x^{a+b} x a ⋅ x b = x a + b 5 8 5 6 = 25 \displaystyle\frac{5^8}{5^6} = 25 5 6 5 8 = 25 x a ⋅ y a = ( x y ) a x^a \cdot y^a = (xy)^a x a ⋅ y a = ( x y ) a ( 3 x ) 4 = 81 x 4 (3x)^4 = 81x^4 ( 3 x ) 4 = 81 x 4 15 = 3 5 \sqrt{15} = \sqrt 3 \sqrt 5 15 = 3 5 ( x a ) b = x a b \left(x^a\right)^b = x^{ab} ( x a ) b = x ab x y a = x ( y a ) x^{y^a} = x^{\left(y^a\right)} x y a = x ( y a ) log x 1 = 0 \log_x 1 = 0 log x 1 = 0 ( a + b ) ( c + d ) = a c + a d + b c + b d (a + b)(c + d) = ac + ad + bc + bd ( a + b ) ( c + d ) = a c + a d + b c + b d expansion, “FOIL” ( a + b ) ( a − b ) = a 2 − b 2 (a + b)(a - b) = a^2 - b^2 ( a + b ) ( a − b ) = a 2 − b 2 difference of squares a b + x = a ( b − x ) b 2 − x \displaystyle\frac{a}{b + \sqrt x} = \frac{a (b - \sqrt x)}{b^2 - x} b + x a = b 2 − x a ( b − x ) (HL) multiplying by the conjugate a 3 ± b 3 = ( a ± b ) ( a 2 ∓ a b + b 2 ) a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2) a 3 ± b 3 = ( a ± b ) ( a 2 ∓ ab + b 2 ) sum and difference of cubes real rate = nominal rate − inflation rate {\text{real rate} = \text{nominal rate} - \text{inflation rate}} real rate = nominal rate − inflation rate an approximation for low rates n C r = n C n − r ^n\C_r = \,^n\C_{n-r} n C r = n C n − r n C r + n C r + 1 = n + 1 C r + 1 ^n\C_r + \,^n\C_{r+1} = \,^{n+1}\C_{r+1} n C r + n C r + 1 = n + 1 C r + 1 for producing Pascal’s triangle by hand
Functions and Calculus formula / property description gradients of perpendicular lines multiply to − 1 -1 − 1 except horizontal and vertical lines x = − d c , y = a c \displaystyle x = -\frac dc, \quad y = \frac ac x = − c d , y = c a asymptotes for f ( x ) = a x + b c x + d \displaystyle f(x) = \frac{ax + b}{cx + d} f ( x ) = c x + d a x + b ( f ∘ g ) ∘ h = f ∘ ( g ∘ h ) (f \circ g) \circ h = f \circ (g \circ h) ( f ∘ g ) ∘ h = f ∘ ( g ∘ h ) associativity of composite functions a = b ⟹ f ( a ) = f ( b ) a = b \implies f(a) = f(b) a = b ⟹ f ( a ) = f ( b ) when f f f a = b ⟺ f ( a ) = f ( b ) a = b \iff f(a) = f(b) a = b ⟺ f ( a ) = f ( b ) when f f f ∣ f ( x ) ∣ > ∣ g ( x ) ∣ ⟺ f ( x ) 2 > g ( x ) 2 \lvert f(x)\rvert > \lvert g(x)\rvert \iff f(x)^2 > g(x)^2 ∣ f ( x )∣ > ∣ g ( x )∣ ⟺ f ( x ) 2 > g ( x ) 2 (HL) useful when solving by hand ( f ∘ f − 1 ) ( x ) = ( f − 1 ∘ f ) ( x ) = x {(f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x} ( f ∘ f − 1 ) ( x ) = ( f − 1 ∘ f ) ( x ) = x when f f f ( f − 1 ) − 1 = f \left(f^{-1}\right)^{-1} = f ( f − 1 ) − 1 = f inverse of inverse is the original function p ( x ) = ( x − a ) q ( x ) + p ( a ) p(x) = (x - a)q(x) + p(a) p ( x ) = ( x − a ) q ( x ) + p ( a ) (HL) remainder theorem, where polynomial p ( x ) p(x) p ( x ) x − a {x - a} x − a f ( − x ) = − f ( x ) f(-x) = -f(x) f ( − x ) = − f ( x ) x x x (HL) odd function, symmetric about the origin f ( − x ) = f ( x ) f(-x) = f(x) f ( − x ) = f ( x ) x x x (HL) even function, symmetric about x = 0 {x = 0} x = 0 increasing if f ′ ( x ) > 0 {f^\prime(x) > 0} f ′ ( x ) > 0 in IB, this means strictly increasing decreasing if f ′ ( x ) < 0 {f^\prime(x) < 0} f ′ ( x ) < 0 in IB, this means strictly decreasing if f f f f ( a ) > f ( b ) ⟺ a > b f(a) > f(b) \iff a > b f ( a ) > f ( b ) ⟺ a > b if f f f f ( a ) > f ( b ) ⟺ a < b f(a) > f(b) \iff a < b f ( a ) > f ( b ) ⟺ a < b eg, − 2 k ≤ 6 -2k \leq 6 − 2 k ≤ 6 k ≥ − 3 k \geq -3 k ≥ − 3 concave up if f ′ ′ ( x ) > 0 f^{\prime\prime}(x) > 0 f ′′ ( x ) > 0 ∪ \cup ∪ concave down if f ′ ′ ( x ) < 0 f^{\prime\prime}(x) < 0 f ′′ ( x ) < 0 ∩ \cap ∩ ave speed = total distance total time \displaystyle\text{ave speed} = \frac{\text{total distance}}{\text{total time}} ave speed = total time total distance
d d x ( y e ∫ P ( x ) d x ) = Q ( x ) e ∫ P ( x ) d x {\displaystyle \frac{\d }{\d x}\left(y\e^{\int P(x)\d x}\right) = Q(x)\e^{\int P(x)\d x}} d x d ( y e ∫ P ( x ) d x ) = Q ( x ) e ∫ P ( x ) d x
(HL) equivalent DE for integrating factor A = ∫ a b g ( x ) − f ( x ) d x \displaystyle {A = \int_a^b g(x) - f(x) \d x} A = ∫ a b g ( x ) − f ( x ) d x (HL) area between functions, g ( x ) > f ( x ) g(x) > f(x) g ( x ) > f ( x ) a < x < b {a < x < b} a < x < b V = π ∫ a b g ( x ) 2 − f ( x ) 2 d x \displaystyle {V = \pi\int_a^b g(x)^2 - f(x)^2 \d x} V = π ∫ a b g ( x ) 2 − f ( x ) 2 d x (HL) volume between surfaces of revolution about x x x g ( x ) > f ( x ) {g(x) > f(x)} g ( x ) > f ( x ) a < x < b {a < x < b} a < x < b
Suppose ∫ f ( x ) d x = F ( x ) + C \displaystyle \int f(x) \d x = F(x) + C ∫ f ( x ) d x = F ( x ) + C
formula description ∫ f ( a x + b ) d x = 1 a F ( a x + b ) + C \displaystyle \int f(ax + b) \d x = \frac{1}{a}F(ax + b) + C ∫ f ( a x + b ) d x = a 1 F ( a x + b ) + C ∫ f ( g ( x ) ) ⋅ g ′ ( x ) d x = F ( g ( x ) ) + C {\displaystyle {\int f(g(x)) \cdot g^\prime(x) \d x} = F(g(x)) + C} ∫ f ( g ( x )) ⋅ g ′ ( x ) d x = F ( g ( x )) + C reverse chain rule y = y 0 + ∫ x 0 x f ( w ) d w \displaystyle y = y_0 + {\int_{x_0}^x f(w) \d w} y = y 0 + ∫ x 0 x f ( w ) d w solution to d y d x = f ( x ) \frac{\d y}{\d x} = f(x) d x d y = f ( x ) ( x 0 , y 0 ) (x_0, y_0) ( x 0 , y 0 )
This final formula removes the need to find the constant of integration when using a calculator. w w w x x x
Trigonometry sin θ = opposite hypotenuse \displaystyle\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} sin θ = hypotenuse opposite
cos θ = adjacent hypotenuse \displaystyle\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} cos θ = hypotenuse adjacent
tan θ = opposite adjacent \displaystyle\tan\theta = \frac{\text{opposite}}{\text{adjacent}} tan θ = adjacent opposite
θ \theta θ sin θ \sin\theta sin θ cos θ \cos\theta cos θ tan θ \tan\theta tan θ 0 ° , 0 0\degree, 0 0° , 0 0 0 0 4 2 = 1 \displaystyle \frac{\sqrt4}{2} = 1 2 4 = 1 0 0 0 30 ° , π 6 \displaystyle 30\degree, \frac{\pi}{6}\quad\quad 30° , 6 π 1 2 \displaystyle \frac{1}{2} 2 1 3 2 \displaystyle \frac{\sqrt3}{2} 2 3 3 3 \displaystyle \frac{\sqrt3}{3} 3 3 45 ° , π 4 \displaystyle 45\degree, \frac{\pi}{4} 45° , 4 π 2 2 \displaystyle \frac{\sqrt{2}}{2} 2 2 2 2 \displaystyle \frac{\sqrt2}{2} 2 2 1 1 1 60 ° , π 3 \displaystyle 60\degree, \frac{\pi}{3} 60° , 3 π 3 2 \displaystyle \frac{\sqrt3}{2} 2 3 1 2 \displaystyle \frac{1}{2} 2 1 3 \sqrt3 3 90 ° , π 2 \displaystyle 90\degree, \frac{\pi}{2} 90° , 2 π 4 2 = 1 \displaystyle \frac{\sqrt4}{2} = 1 2 4 = 1 0 0 0 undefined
Sum of angles in a triangle is π \pi π 180 ° 180 \degree 180°
The two shorter sides of a triangle must sum to longer than the third.
Model: y = a sin ( b ( x − c ) ) + d y = a \sin (b(x-c)) + d \quad y = a sin ( b ( x − c )) + d cos \cos cos
parameter description ∣ a ∣ = max − min 2 \displaystyle \lvert a \rvert = \frac{\text{max} - \text{min}}{2} ∣ a ∣ = 2 max − min a < 0 a<0 a < 0 b = 2 π period \displaystyle b = \frac{2\pi}{\text{period}} b = period 2 π or 360 ° period \frac{360\degree}{\text{period}} period 360° d = max + min 2 \displaystyle d = \frac{\text{max} + \text{min}}{2} d = 2 max + min
identity description sin ( π 2 − θ ) = cos θ \sin \left(\frac\pi2 - \theta\right) = \cos \theta sin ( 2 π − θ ) = cos θ sin ( π − θ ) = sin θ \sin(\pi - \theta) = \sin \theta sin ( π − θ ) = sin θ HL cos ( π − θ ) = − cos θ \cos(\pi - \theta) = -\cos \theta cos ( π − θ ) = − cos θ HL tan ( π − θ ) = − tan θ \tan(\pi - \theta) = -\tan \theta tan ( π − θ ) = − tan θ HL
trig equation solutions for θ \theta θ sin θ = k \sin \theta = k sin θ = k sin − 1 k + 2 n π − sin − 1 k + ( 2 n + 1 ) π \sin^{-1} k + 2n\pi \\ -\sin^{-1} k + (2n + 1)\pi sin − 1 k + 2 nπ − sin − 1 k + ( 2 n + 1 ) π cos θ = k \cos \theta = k cos θ = k ± cos − 1 k + 2 n π \pm\cos^{-1} k + 2n\pi ± cos − 1 k + 2 nπ tan θ = k \tan \theta = k tan θ = k tan − 1 k + n π \tan^{-1} k + n\pi tan − 1 k + nπ
The line y = x tan θ y = x \tan \theta y = x tan θ θ \theta θ x x x tan θ \tan \theta tan θ
Probability and Statistics formula / property description an outlier is at least 1.5 IQR from nearest quartile × \times × P ( μ − σ < X < μ + σ ) ≈ 0.68 \mathrm P(\mu - \sigma < X < \mu + \sigma) \approx 0.68 P ( μ − σ < X < μ + σ ) ≈ 0.68 normal distribution, 1 stdev from mean on both sides P ( μ − 2 σ < X < μ + 2 σ ) ≈ 0.95 \mathrm P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 0.95 P ( μ − 2 σ < X < μ + 2 σ ) ≈ 0.95 normal distribution, 2 stdev from mean on both sides P ( μ − 3 σ < X < μ + 3 σ ) ≈ 0.997 {\mathrm P(\mu - 3\sigma < X < \mu + 3\sigma) \approx 0.997} P ( μ − 3 σ < X < μ + 3 σ ) ≈ 0.997 normal distribution, 3 stdev from mean on both sides ( x ˉ , y ˉ ) (\bar x, \bar y) ( x ˉ , y ˉ ) y y y x x x x x x y y y
Counting Principles (HL) Let M , N ∈ Z + M, N \in \mathbb Z^+ M , N ∈ Z + counting notes and counting tutorial .
total options description M ⋅ N M \cdot N M ⋅ N N N N M M M M d \frac M d d M if each arrangement appears d d d M d \frac Md d M c d c^d c d c c c d d d M + N M + N M + N union of cases or disjoint sets n − m + 1 , m ≤ n {n - m + 1, \; m \leq n} n − m + 1 , m ≤ n number of integers from m m m n n n M C m N C n {^M\C_m \;^N\C_n} M C m N C n choose exactly m m m M M M n n n N N N M + N M + N M + N
Complex Numbers (HL) formula / property description ∣ a + b i ∣ = r = a 2 + b 2 \lvert a + b\i\rvert = r = \sqrt{a^2 + b^2} ∣ a + b i ∣ = r = a 2 + b 2 modulus arg z = { undefined z = 0 π 2 a = 0 , b > 0 − π 2 a = 0 , b < 0 arctan b a a > 0 arctan b a + π a < 0 , b ≥ 0 arctan b a − π a < 0 , b < 0 \displaystyle
\arg \, z= \begin{cases} \text{undefined} &z = 0 \\
\frac{\pi}{2} &a = 0, b > 0 \\
-\frac{\pi}{2} &a = 0, b < 0 \\
\arctan{\frac ba} &a > 0 \\
\arctan{\frac ba} + \pi &a < 0, b \geq 0 \\
\arctan{\frac ba} - \pi &a < 0, b < 0 \end{cases} arg z = ⎩ ⎨ ⎧ undefined 2 π − 2 π arctan a b arctan a b + π arctan a b − π z = 0 a = 0 , b > 0 a = 0 , b < 0 a > 0 a < 0 , b ≥ 0 a < 0 , b < 0
arg z \arg \, z arg z ] − π , π ] {]-\pi, \pi]} ] − π , π ] [ 0 , 2 π [ {[0, 2\pi[} [ 0 , 2 π [ z ∗ = a − b i = r c i s ( − θ ) {z^* = a - bi = r\cis(-\theta)} z ∗ = a − bi = r cis ( − θ ) complex conjugate of z = a + b i = r c i s ( θ ) z = a + bi = {r\cis(\theta)} z = a + bi = r cis ( θ ) ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i {(a + b\i) + (c + d\i) = (a + c) + (b + d)\i} ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i
addition in Cartesian form ( a + b i ) ( c + d i ) = a c − b d + ( a d + b c ) i {(a + b\i)(c + d\i) = ac - bd + (ad + bc)\i} ( a + b i ) ( c + d i ) = a c − b d + ( a d + b c ) i
multiplication in Cartesian form 1 a + b i = a − b i a 2 + b 2 \displaystyle \frac{1}{a + b\i} = \frac{a - b\i}{a^2 + b^2} a + b i 1 = a 2 + b 2 a − b i 1 z = z ∗ ∣ z ∣ 2 \displaystyle {\frac{1}{z} = \frac{z^*}{\lvert z\rvert^2}} z 1 = ∣ z ∣ 2 z ∗
division in Cartesian form ( r 1 c i s θ 1 ) ⋅ ( r 2 c i s θ 2 ) = r 1 r 2 c i s ( θ 1 + θ 2 ) {\left(r_1 \cis \theta_1\right)\cdot\left(r_2 \cis \theta_2\right) = r_1r_2 \cis \left(\theta_1 + \theta_2\right)} ( r 1 cis θ 1 ) ⋅ ( r 2 cis θ 2 ) = r 1 r 2 cis ( θ 1 + θ 2 )
multiplication in polar form 1 r c i s θ = 1 r c i s ( − θ ) \displaystyle {\frac{1}{r \cis \theta} = \frac1r \cis \left(-\theta\right)} r cis θ 1 = r 1 cis ( − θ )
division in polar form z k = r 1 n c i s ( θ + 2 k π n ) \displaystyle z_k = r^{\frac1n} \cis \left(\frac{\theta + 2k\pi}{n}\right) z k = r n 1 cis ( n θ + 2 k π )
roots to z n = r c i s θ , k = 0 , 1 , … , n − 1 {z^n = r \cis \theta, \ k = 0, 1, \dots, n-1} z n = r cis θ , k = 0 , 1 , … , n − 1
Vectors (HL) With vectors v \mathbf v v w \mathbf w w ∣ v ∣ \lvert \mathbf v\rvert ∣ v ∣ ∣ w ∣ \lvert \mathbf w\rvert ∣ w ∣
formula / property description k v = < k v 1 , k v 2 , k v 3 > k \mathbf v = \lt kv_1, kv_2, kv_3 \gt k v =< k v 1 , k v 2 , k v 3 > scalar multiplication ∣ v ⋅ w ∣ = ∣ v ∣ ∣ w ∣ \lvert \mathbf v \cdot \mathbf w \rvert = \lvert \mathbf v \rvert \lvert \mathbf w \rvert ∣ v ⋅ w ∣ = ∣ v ∣ ∣ w ∣ parallel vectors ∣ v ⋅ w ∣ = 0 \lvert \mathbf v \cdot \mathbf w \rvert = 0 ∣ v ⋅ w ∣ = 0 perpendicular vectors v ⋅ w = w ⋅ v \mathbf v \cdot \mathbf w = \mathbf w \cdot \mathbf v v ⋅ w = w ⋅ v v × w = − w × v \mathbf v \times \mathbf w = -\mathbf w \times \mathbf v v × w = − w × v order matters u ( v ⋅ w ) = u ⋅ v + u ⋅ w \mathbf u (\mathbf v \cdot \mathbf w) = \mathbf u \cdot \mathbf v + \mathbf u \cdot \mathbf w u ( v ⋅ w ) = u ⋅ v + u ⋅ w also for cross product ( k v ) ⋅ w = k ( v ⋅ w ) (k \mathbf v) \cdot \mathbf w = k (\mathbf v \cdot \mathbf w) ( k v ) ⋅ w = k ( v ⋅ w ) also for cross product ∣ v ∣ 2 = v ⋅ v \lvert \mathbf v\rvert^2 = \mathbf v \cdot\mathbf v ∣ v ∣ 2 = v ⋅ v cos 0 = 1 \cos 0 = 1 cos 0 = 1 ∣ v + w ∣ 2 = ∣ v ∣ 2 + 2 v ⋅ w + ∣ w ∣ 2 {\lvert \mathbf v + \mathbf w \rvert^2 = \lvert \mathbf v\rvert^2 + 2 \mathbf v \cdot \mathbf w + \lvert \mathbf w\rvert^2} ∣ v + w ∣ 2 = ∣ v ∣ 2 + 2 v ⋅ w + ∣ w ∣ 2 n 1 × n 2 ⋅ n 3 ≠ 0 \bm{n_1} \times \bm{n_2} \cdot \bm{n_3} \neq 0 n 1 × n 2 ⋅ n 3 = 0 condition for a unique intersection of 3 planes s = ∣ a ∣ cos θ = a ⋅ b ∣ b ∣ \displaystyle s = \left\lvert\bm a \right\rvert \cos\theta = \frac{\bm a \cdot \bm b}{\lvert\bm b\rvert} s = ∣ a ∣ cos θ = ∣ b ∣ a ⋅ b distances and scalar projection D = ∣ O P → ⋅ n − d ∣ ∣ n ∣ \displaystyle D = \frac{\left\lvert\overrightarrow{OP}\cdot \bm n - d\right\rvert}{\lvert\bm n\rvert} D = ∣ n ∣ O P ⋅ n − d distance between a point/parallel line and plane D = ∣ d 2 − d 1 ∣ ∣ n ∣ \displaystyle D = \frac{\left\lvert d_2 - d_1\right\rvert}{\lvert\bm n\rvert} D = ∣ n ∣ ∣ d 2 − d 1 ∣ distance between 2 parallel planes D = ∣ ( b − a ) ⋅ ( d 1 × d 2 ) ∣ ∣ d 1 × d 2 ∣ \displaystyle D = \frac{\left\lvert(\bm b - \bm a) \cdot (\bm{d_1} \times \bm{d_2})\right\rvert}{\lvert \bm{d_1} \times \bm{d_2}\rvert} D = ∣ d 1 × d 2 ∣ ∣ ( b − a ) ⋅ ( d 1 × d 2 ) ∣ distance between 2 skew lines D = ∣ ( b − a ) × d ∣ ∣ d ∣ \displaystyle D = \frac{\left\lvert(\bm b - \bm a) \times \bm d\right\rvert}{\lvert \bm d \rvert} D = ∣ d ∣ ∣ ( b − a ) × d ∣ distance between point and line or between 2 parallel lines
While IB accepts directly using the various distance formulas, it may be better to derive them each time using scalar projection.