MU123 Unit 14 Notes

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

The quadrants

    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)