# Mathematics Glossary

ABCDEFGHIJKLMNOPQRSTUVWXYZA Absolute error - The difference between the actual value and the measured value indicated by an instrument. Acount servicing fee - Ongoing account keeping fees. Adjacent side - The side next to the right-angle used for reference in a right-angled triangle. Adjacent sides usually found in triangles and other polygons. Algebraic modelling - When [...]

Let $z = \cos \theta + i \sin \theta$. (a)   By considering the real part of $z^4$, prove $\cos 4\theta = \cos^4 \theta - 6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta$. \begin{align} \displaystyle \require{color} z &= \cos \theta + i \sin \theta \\ z^4 &= (\cos [...] # Adding Multiples of Consecutive Odd Numbers by Mathematical Induction (a) Factorise \( 4x^3 + 18x^2 + 23x + 9. \begin{align} \displaystyle 4x^3 + 18x^2 + 23x + 9 &= 4x^3 + 4x^2 + 14x^2 + 23x + 9 \\ &= 4x^2 (x+1) + 14x^2 + 14x + 9x + 9 \\ &= 4x^2 (x+1) + 14x(x+1) + 9(x+1) \\ &= (x+1)(4x^2 [...] # Mathematical Induction Regarding Factorials Prove by mathematical induction that for al lintegers \( n \ge 1 , $$\dfrac{1}{2!} + \dfrac{2}{3!} + \dfrac{3}{4!} + \cdots + \dfrac{n}{(n+1)!} = 1 - \dfrac{1}{(n+1)!}$$ Step 1: Show it is true for $n=1$. \begin{align} \displaystyle \text{LHS } &= \dfrac{1}{2!} = \dfrac{1}{2} \\ \text{RHS } &= 1 - \dfrac{1}{2!} \\ [...] # Applications of the Unit Circle The identify \cos^2 \theta + \sin^2 \theta = 1 is required for finding trigonometric ratios. Example 1 Find exactly the possible values of \cos \theta for \sin \theta = \dfrac{5}{8}. \( \begin{align} \displaystyle \cos^2 \theta + \sin^2 \theta &= 1 \\ \cos^2 \theta + \sin^2 \dfrac{5}{8} &= 1 \\ \cos^2 \theta + \dfrac{25}{64} &= 1 [...] # Trigonometric Ratios Circles with Cnetre (0,0) Consider a circle with centre (0,0) and radius r units. Suppose (x,y) is any point on this circle. Using ths distance formula; \( \begin{align} \displaystyle \sqrt{(x-0)^2+(y-0)^2} &= r \\ \therefore x^2+y^2 &= r^2 \end{align} $x^2+y^2 = r^2$ is the equation of a circle with centre $(0,0)$ and radius $r$. The [...] # Arc Length and Sector Area

You should know with these terms relating to the parts of a circle. The centre of a circle is the point which is equidistant from all points on the circle. A radius of a circle a straight line joining the centre of a circle to any point on the circumference. A minor arc is an [...] Degree Measurement of Angles One full revolution makes an angle of $360^{\circ}$, and the angle on a straight line is $180^{\circ}$. Therefore, one degree, $1^{\circ}$, can be defined as $\dfrac{1}{360}$ of one full revolution. For greater accuracy we define one minute, $1'$, as $\dfrac{1}{60}$ of one degree and one second, $1^{\prime \prime}$, as $\dfrac{1}{60}$ of [...]