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THE KNOWLEDGE / MATHS

6th Form Further knowledge

Facts currently selected by your teacher · 25 facts · questions and answers

Algebra

01

Discriminant and how it determines the number of roots of a quadratic

Δ=b2−4ac\Delta=b^2-4ac

Δ > 0: two distinct real roots. Δ = 0: one repeated real root. Δ < 0: no real roots.

02

Quadratic formula

x=−b±b2−4ac2ax=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}
03

The factor theorem

(x−a) is a factor of f(x)  ⟺  f(a)=0(x-a)\text{ is a factor of }f(x)\iff f(a)=0

Trigonometry

05

The sine rule

asin⁡A=bsin⁡B=csin⁡C\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}
06

The cosine rule

a2=b2+c2−2bccos⁡Aa^2=b^2+c^2-2bc\cos A
07

Area of a triangle using two sides and the included angle

A=12absin⁡CA=\dfrac12 ab\sin C
10

Sketch the graph of y = sin x in degrees and radians

y = sin x
−1100°π/290°π180°3π/2270°2π360°xy

Period 2π (360°); range [−1, 1].

11

Sketch the graph of y = cos x in degrees and radians

y = cos x
−1100°π/290°π180°3π/2270°2π360°xy

Period 2π (360°); range [−1, 1].

12

Sketch the graph of y = tan x in degrees and radians

y = tan x
00°π/290°π180°3π/2270°2π360°xy

Period π (180°); asymptotes at x = π/2 + kπ, for any integer k.

13

Special values of sin, cos and tan (in both degrees and radians)

θ0°30°45°60°90°
Radians0π/6π/4π/3π/2
sin θ01/2√2/2√3/21
cos θ1√3/2√2/21/20
tan θ01/√31√3undefined

Angles: 0°, 30°, 45°, 60°, 90° (or 0, π/6, π/4, π/3, π/2 radians). tan 90° is undefined.

14

Arc length when the angle is in radians

s=rθs=r\theta
15

Area of a sector when the angle is in radians

A=12r2θA=\dfrac12r^2\theta
16

Identity linking tan θ, sin θ and cos θ

tan⁡θ=sin⁡θcos⁡θ\tan\theta=\dfrac{\sin\theta}{\cos\theta}
56

Small-angle approximations for sin θ, cos θ and tan θ

sin⁡θ≈θ,cos⁡θ≈1−θ22,tan⁡θ≈θ\sin\theta\approx\theta,\qquad\cos\theta\approx1-\dfrac{\theta^2}{2},\qquad\tan\theta\approx\theta

θ is in radians and close to zero.

Coordinate geometry

08

Equation of a circle

Centre (a, b), radius r:

(x−a)2+(y−b)2=r2(x-a)^2+(y-b)^2=r^2
09

Equation of a straight line with gradient m passing through (x₁, y₁)

y−y1=m(x−x1)y-y_1=m(x-x_1)

Graphs

24

Sketch the graph of y = x²

y = x²
-2-1012(0, 0)xy
25

Sketch the graph of y = x³

y = x³
-2-1012(0, 0)xy
26

Sketch the graph of y = 1/x

y = 1/x
-2-10123xy

Neither axis is an intercept; both axes are asymptotes.

30

Transformation y = f(x) + a

Translate by (0a)\text{Translate by }\begin{pmatrix}0\\a\end{pmatrix}
31

Transformation y = f(x + a)

Translate by (−a0)\text{Translate by }\begin{pmatrix}-a\\0\end{pmatrix}
32

Transformation y = af(x)

Stretch parallel to the y-axis by scale factor a.

33

Transformation y = f(ax)

Stretch parallel to the x-axis by scale factor 1/a (for a > 0).

34

Transformation y = −f(x)

Reflect in the x-axis.

35

Transformation y = f(−x)

Reflect in the y-axis.