# 1 Communicating clearly

## 1.1 Interpreting information about risk

Probability, chance or risk defines how likely is something to occur.

Probabilities can be expressed as fractions, decimals or percentages, or in the form of an $x$ in $y$ chance.

## 1.2 Interpreting graphs and charts

Comparative bar chart is a bar chart with tow or more bars for each data item. Pictogram is a chart that conveys numerical information by means of pictures.

# 2 Extending your mathematical skills

## 2.1 Developing your equation-solving skills

### Using shortcuts

#### Cross-multiplying

$$\frac{A}{B} = \frac{C}{D} \\[1em] BD \times \frac{A}{B} = BD \times \frac{C}{D} \\[1em] AD = BC$$

## 2.2 Solving trigonometric equations

In general, for any acute angle $\theta$ the for angles

$\theta$, $180^\circ-\theta$, $180^\circ+\theta$, $360^\circ-\theta$

all have the same sine, cosine and tangent, except for the signs. The signs are determined by the sings that th $x$- and $y$-coordinates take in the different quadrats:

• In the first quadrant, $x$ and $y$ are both positive, so sine, cosine and tangent are all positive.
• In the second quadrant, $x$ is negative and $y$ is positive, so sine is positive and cosine is negative, and hence tangent is negative.
• In the third quadrant, $x$ and $y$ are both negative, so sine and cosine are both negative, and hence tangent is positive.
• In the fourth quadrant, $x$ is positive and $y$ is negative, so sine is negative and cosine is positive, and hence tangent is negative.

The sings can be remembered using CAST:

S (Sin positive)A (All positive)
T (Tan positive)C (Cos positive)

## 2.3 Finding the angle of inclination of a line

Angle of inclination of a line is its angle measured anticlockwise from the positive direction of the $x$-axis, when the line is drawn on a pair of axes with equal scales.

### Gradient and angle of inclination of a straight line

For any straight line with angle of inclination $\theta$, $$\text{gradient} = \tan \theta$$ Remember that the angle of inclination is measured when the line is drawn on axes with equal scales.

## 2.4 Solving trigonometric equations in radians

    pi/2
|
pi ----- 0 and 2pi
|
3pi/2



## 4.2 Exploring sinusoidal functions

A curve that can be obtained from the graph of the sine function by shifting, stretching or compressing it horizontally or vertically is called a sinusoidal curve. A function whose graph is sinusoidal curve is called a sinusoidal function.

General sine functions and general cosine functions are functions with roles of following forms: $$y = a \sin(b(x - c)) + d$$ and $$y = a \cos (b(x + c)) + d$$ where $a$, $b$, $c$ and $d$ are constants.

The $y$-values of any sinusoidal function oscillates between a minimum value and a maximum value and the shape of the graph is continually repeated. Period of the graph is the length on the $x$-axis that it takes for the graph to repeat. Any section of the graph covering this length on the $x$-axis is referred to as an oscillation.

### The period of general sine or cosine function

The period of $y = a \sin(b(x - c)) + d$ or $y = a \cos (b(x + c)) + d$ is: $$\frac{2\pi}{|b|}$$

### The graph of a general sine function

The graph of the equation $$y = a \sin(b(x - c)) + d$$ where $a$ and $b$ are positive, and $c$ and $d$ can take any values, has the following features.

• $a$ is the amplitude: the distance between the center line and the maximum (or minimum) value.
• $b$ determines the period, which is equal to $2\pi/b$.
• $c$ is the horizontal displacement: the amount that the graph of $y = a \sin (bx) + d$ is shifhted to the right to obtain the graph of $y = a \sin (b(x - c)) + d$. (The shift is to the left if $c$ is negative.)
• $d$ is the vertical displacement: the amount that the center line is shifted up from the $x$-axis. (The shift is down if $d$ is negative.)
• The minimum value is $d - a$ (the vertical displacement minus the amplitude)
• The maximum value is $d + a$ (the vertical displacement plus the amplitude)