Use the identities sin²A + cos²A = 1, sec²A = 1 + tan²A and cosec²A = 1 + cot²A

IGCSE Additional Mathematics (0606) – Summary Notes

Learning Objectives

• Manipulate functions, quadratics, polynomials and equations.
• Apply logarithmic and exponential laws.
• Analyse straight‑line graphs and the geometry of circles.
• Use the fundamental trigonometric identities to simplify, solve and prove further identities.

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1. Functions

1.1 Key Concepts

• Definition: A function $f$ assigns to each element $x$ in a domain a unique element $y$ in a range, written $y=f(x)$.
• Domain & Range: The set of admissible $x$‑values and the corresponding $y$‑values.
• One‑to‑One (Injective): $f(a)=f(b)\Rightarrow a=b$. This condition is required for an inverse function to exist.
• Existence of an Inverse:
  • The function must be one‑to‑one on the considered domain.
  • If necessary, restrict the domain so that the graph is monotonic.
• Inverse Function: $f^{-1}(y)=x$ such that $f(x)=y$. Graphically it is the reflection of $y=f(x)$ in the line $y=x$.
• Composition: $(f\circ g)(x)=f\bigl(g(x)\bigr)$. The notation $fg(x)$ is *not* used in the syllabus – always write $f\circ g$ or $(f\circ g)(x)$.
• Notation Checklist:

1.2 Sketching a Function and Its Inverse

1. Find intercepts, asymptotes and intervals of monotonicity.
2. Draw the curve of $y=f(x)$.
3. Reflect every point $(a,b)$ in the line $y=x$ to obtain $(b,a)$ – this gives the graph of $y=f^{-1}(x)$.

1.3 Worked Example – Inverse of a Non‑Linear Function

Find the inverse of $f(x)=\sqrt{x-1}$ and state the domain of $f^{-1}$.

1. Write $y=\sqrt{x-1}$ and square: $y^{2}=x-1\;\Longrightarrow\;x=y^{2}+1$.
2. Interchange $x$ and $y$: $y=x^{2}+1$.
3. Hence $f^{-1}(x)=x^{2}+1$, with domain $x\ge0$ (because the original $y$ was $\ge0$).

1.4 Worked Example – Composition

Given $f(x)=\dfrac{1}{x}$ and $g(x)=2x+3$, find $(f\circ g)(x)$ and $(g\circ f)(x)$.

• $(f\circ g)(x)=f\bigl(g(x)\bigr)=\dfrac{1}{2x+3}$.
• $(g\circ f)(x)=g\bigl(f(x)\bigr)=2\!\left(\dfrac{1}{x}\right)+3=\dfrac{2}{x}+3$.

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2. Quadratic Functions

2.1 Standard and Vertex Forms

Form	Expression	Key Parameters
Standard	$ax^{2}+bx+c$	$aeq0$
Vertex	$a(x-h)^{2}+k$	Vertex $(h,k)$

2.2 Completing the Square

Convert $ax^{2}+bx+c$ to vertex form:

\[ ax^{2}+bx+c = a\Bigl(x^{2}+\frac{b}{a}x\Bigr)+c = a\Bigl[\Bigl(x+\frac{b}{2a}\Bigr)^{2}-\Bigl(\frac{b}{2a}\Bigr)^{2}\Bigr]+c = a\Bigl(x+\frac{b}{2a}\Bigr)^{2}-\frac{b^{2}}{4a}+c. \]

2.3 Discriminant

\[ \Delta=b^{2}-4ac \]

• $\Delta>0$ – two distinct real roots.
• $\Delta=0$ – one repeated real root (the parabola touches the $x$‑axis).
• $\Delta<0$ – no real roots (the parabola lies wholly above or below the $x$‑axis).

2.4 Worked Example – Solving a Quadratic Inequality

Solve $x^{2}-5x+6\le0$.

1. Factor: $(x-2)(x-3)\le0$.
2. Critical points $x=2,\,3$. Test the three intervals:
   • $x<2$: product $>0$.
   • $2\le x\le3$: product $\le0$.
   • $x>3$: product $>0$.
3. Solution: $\boxed{2\le x\le3}$.

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3. Factors of Polynomials

3.1 Remainder & Factor Theorems

• Remainder theorem: When $f(x)$ is divided by $(x-a)$, the remainder is $f(a)$.
• Factor theorem: $(x-a)$ is a factor of $f(x)$ iff $f(a)=0$.

3.2 Synthetic Division (example)

Find a factor of $f(x)=2x^{3}-3x^{2}-8x+12$ given that $x=2$ is a root.

2 | 2  -3  -8  12
    4   2  -12
  ----------------
    2   1  -6   0

Result: $f(x)=(x-2)(2x^{2}+x-6)$. The quadratic further factors as $(2x-3)(x+2)$.

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4. Equations & Inequalities

4.1 Linear Equations

General form $ax+b=cx+d\;\Longrightarrow\;(a-c)x=b-d$.

4.2 Absolute‑Value Equations

• $|ax+b|=c\;(c\ge0)\;\Longrightarrow\;ax+b=c\;$ or $\;ax+b=-c$.
• $|ax+b|=cx+d$ – solve by considering the sign of $cx+d$; if necessary square both sides.

4.3 Worked Example – Solving $|2x-5|=x+1$

1. Case 1: $2x-5\ge0\;(x\ge2.5)$. Equation becomes $2x-5=x+1\Rightarrow x=6$ (valid).
2. Case 2: $2x-5<0\;(x<2.5)$. Equation becomes $-(2x-5)=x+1\Rightarrow -2x+5=x+1\Rightarrow3x=4\Rightarrow x=\tfrac43$ (valid).
3. Solution: $\boxed{x=\tfrac43\text{ or }x=6}$.

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5. Simultaneous Equations

5.1 Linear–Linear System

Example: $\begin{cases}3x+2y=7\\5x-y=4\end{cases}$

1. Multiply the second equation by 2: $10x-2y=8$.
2. Add to the first: $13x=15\;\Rightarrow\;x=\dfrac{15}{13}$.
3. Substitute back: $5\!\left(\dfrac{15}{13}\right)-y=4\;\Rightarrow\;y=\dfrac{75}{13}-4=\dfrac{23}{13}$.

5.2 Linear–Quadratic System (example)

Solve $\begin{cases}y=2x+1\\y^{2}=9x\end{cases}$.

1. Substitute $y$: $(2x+1)^{2}=9x\;\Longrightarrow\;4x^{2}+4x+1-9x=0\Rightarrow4x^{2}-5x+1=0$.
2. Quadratic formula: $x=\dfrac{5\pm\sqrt{25-16}}{8}=\dfrac{5\pm3}{8}$ → $x=1$ or $x=\dfrac14$.
3. Corresponding $y$: $y=2(1)+1=3$ and $y=2\!\left(\dfrac14\right)+1=\dfrac32$.
4. Solutions: $(1,3)$ and $\left(\dfrac14,\dfrac32\right)$.

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6. Logarithmic & Exponential Functions

6.1 Definitions

• Exponential: $y=a^{x}$, $a>0$, $aeq1$.
• Logarithm: $x=\log_{a}y\;\Longleftrightarrow\;a^{x}=y$.

6.2 Laws of Indices

\[ a^{m}a^{n}=a^{m+n},\qquad \frac{a^{m}}{a^{n}}=a^{m-n},\qquad (a^{m})^{n}=a^{mn}. \]

6.3 Laws of Logarithms

\[ \log_{a}(mn)=\log_{a}m+\log_{a}n,\qquad \log_{a}\!\left(\frac{m}{n}\right)=\log_{a}m-\log_{a}n,\qquad \log_{a}(m^{n})=n\log_{a}m. \]

6.4 Solving $a^{x}=b$

Take logarithms of any convenient base:

\[ x=\frac{\log b}{\log a}\qquad\text{(or }\displaystyle x=\frac{\ln b}{\ln a}\text{)}. \]

Example: $3^{x}=81\;\Longrightarrow\;x=\dfrac{\log81}{\log3}=4$.

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7. Straight‑Line Graphs

7.1 Equation Forms

• Standard form: $y=mx+c$ (gradient $m$, $y$‑intercept $c$).
• Two‑point form: $m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}$.

7.2 Parallel & Perpendicular Lines

• Parallel: $m_{1}=m_{2}$.
• Perpendicular: $m_{1}m_{2}=-1$ (provided neither line is vertical).

7.3 Log‑Log Straight Line

If $y=Ax^{n}$, then $\log y=n\log x+\log A$. On a log–log plot the points lie on a straight line with gradient $n$ and intercept $\log A$.

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8. Coordinate Geometry of the Circle

8.1 Standard Form

\[ (x-h)^{2}+(y-k)^{2}=r^{2} \]

• Centre $C(h,k)$, radius $r$.

8.2 Intersection with a Straight Line

Replace $y$ (or $x$) in the circle equation by the line equation $y=mx+c$ (or $x=const$). The resulting quadratic in $x$ (or $y$) gives:

• Two distinct real roots – the line cuts the circle at two points.
• One repeated root – the line is tangent.
• No real root – the line does not meet the circle.

8.3 Tangent at a Point $(x_{1},y_{1})$ on the Circle

\[ (x_{1}-h)(x-h)+(y_{1}-k)(y-k)=r^{2} \]

or, using gradient form,

\[ y-y_{1}= -\frac{x_{1}-h}{\,y_{1}-k\,}\;(x-x_{1}). \]

8.4 Two‑Circle Problems

• If the circles have equal radii, the line joining their centres is the perpendicular bisector of the common chord.
• In the general case, subtract the two circle equations to obtain the equation of the common chord (a straight line), then solve this line with either circle to find the intersection points.

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9. Trigonometry – Fundamental Identities

9.1 Core Pythagorean Identities

\[ \sin^{2}A+\cos^{2}A=1\qquad \sec^{2}A=1+\tan^{2}A\qquad \csc^{2}A=1+\cot^{2}A \]

9.2 Deriving the Secant–Tangent and Cosecant–Cotangent Forms

1. Divide $\sin^{2}A+\cos^{2}A=1$ by $\cos^{2}A$: \[ \frac{\sin^{2}A}{\cos^{2}A}+1=\frac{1}{\cos^{2}A} \Longrightarrow \tan^{2}A+1=\sec^{2}A. \]
2. Divide the same identity by $\sin^{2}A$: \[ 1+\frac{\cos^{2}A}{\sin^{2}A}=\frac{1}{\sin^{2}A} \Longrightarrow 1+\cot^{2}A=\csc^{2}A. \]

9.3 Additional Useful Identities

• $\displaystyle \tan A=\frac{\sin A}{\cos A},\qquad \cot A=\frac{\cos A}{\sin A}$
• $\displaystyle \sec A=\frac{1}{\cos A},\qquad \csc A=\frac{1}{\sin A}$
• Double‑angle: $\displaystyle \cos2A=1-2\sin^{2}A=2\cos^{2}A-1$
• Half‑angle: $\displaystyle \sin^{2}A=\frac{1-\cos2A}{2},\quad \cos^{2}A=\frac{1+\cos2A}{2}$

9.4 Worked Example 1 – Simplify an Expression

Simplify $\displaystyle \frac{1}{\sec^{2}\theta}+\tan^{2}\theta$.

1. Recall $\displaystyle \frac{1}{\sec^{2}\theta}=\cos^{2}\theta$.
2. Use the identity $\tan^{2}\theta= \sec^{2}\theta-1$.
3. Substitute: \[ \frac{1}{\sec^{2}\theta}+\tan^{2}\theta =\cos^{2}\theta+(\sec^{2}\theta-1) =\cos^{2}\theta+\frac{1}{\cos^{2}\theta}-1. \]
4. Combine over a common denominator: \[ =\frac{\cos^{4}\theta+1-\cos^{2}\theta}{\cos^{2}\theta} =\frac{(\cos^{2}\theta-1)^{2}+2\cos^{2}\theta}{\cos^{2}\theta} =\frac{\sin^{4}\theta+2\cos^{2}\theta}{\cos^{2}\theta}. \] (Any equivalent simplified form is acceptable; a common answer is $\displaystyle \frac{1}{\cos^{2}\theta}-\sin^{2}\theta$.)

9.5 Worked Example 2 – Solving a Trigonometric Equation

Solve $\displaystyle \sec^{2}x-\tan^{2}x=1$ for $0^\circ\le x<360^\circ$.

1. Use the identity $\sec^{2}x=1+\tan^{2}x$.
2. Substitute: $(1+\tan^{2}x)-\tan^{2}x=1\;\Longrightarrow\;1=1$, which is always true.
3. Therefore the equation holds for every $x$ where $\sec x$ and $\tan x$ are defined, i.e. $xeq90^\circ,270^\circ$.
4. Solution set: $0^\circ\le x<360^\circ$, $xeq90^\circ,270^\circ$.

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10. Quick Reference – Symbols & Conventions

• Angles are measured in degrees unless the symbol “rad” is explicitly used.
• All trigonometric functions refer to acute angles in right‑triangle contexts, but the identities are valid for any angle in the unit‑circle sense.
• When writing powers of trig functions, use parentheses for clarity: $\sin^{2}\!A=(\sin A)^{2}$.
• For logarithms, $\log$ without a subscript denotes base 10; $\ln$ denotes natural logarithm (base $e$).