Mathematics Module Review: Trigonometry, Bearings, Matrices, and Univariate Data Analysis

Vocabulary and definition flashcards covering Trigonometry, Bearings, Matrices, and Univariate Data Analysis.

Cosine Ratio Formula

In a right-angled triangle, $\text{cos}(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

Finding an Unknown Side Length (Example)

Solving for $x$ when $\text{sin}(30^\text{o}) = \frac{x}{12}$ gives $x = 12 \times \text{sin}(30^\text{o}) = 12 \times 0.5 = 6$

Finding an Unknown Angle (Example)

The inverse trigonometric function used to find $\theta$ when $\text{tan}(\theta) = 1$ is $\theta = \text{tan}^{-1}(1) = 45^\text{o}$

Area of a Triangle Using Trigonometry

Formula for the area of a triangle given two sides $a$, $b$ and included angle $C$: $\text{Area} = \frac{1}{2}ab\text{sin}(C)$

Heron's Formula

$\text{Area} = \text{sqrt}(s(s - a)(s - b)(s - c))$, where the semi-perimeter $s = \frac{a + b + c}{2}$ for a triangle with sides $a$, $b$, and $c$

The Sine Rule

Formula used to calculate an unknown side length or angle: $\frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)}$

The Cosine Rule for Angles

Formula to find an unknown angle $C$: $\text{cos}(C) = \frac{a^2 + b^2 - c^2}{2ab}$

Back Bearing Calculation

If the true bearing of B from A is $135^\text{o}\text{T}$, the back bearing of A from B is $315^\text{o}\text{T}$ ($135^\text{o} + 180^\text{o}$)

Angles of Elevation and Depression

The angle of elevation from point A to point B is equal to the angle of depression from point B to point A because they are alternate interior angles between parallel horizontal lines.

Straight-line Displacement Example

A person walking $3\text{ km}$ East and $4\text{ km}$ North has a straight-line displacement of $5\text{ km}$ ($\text{sqrt}(3^2 + 4^2) = \text{sqrt}(25) = 5$)

Matrix Dimensions Definition

Defined as $\text{Rows} \times \text{Columns}$ ($m \times n$)

Matrix Multiplication Condition

To multiply matrix $A$ ($m \times n$) by matrix $B$ ($p \times q$), the inner dimensions must be equal ($n = p$). The resulting matrix will have dimensions $m \times q$.

Matrices for Costing and Pricing

To find total cost by multiplying a quantity matrix by a price matrix, if quantities are in a row matrix ($1 \times n$), prices must be arranged in a column matrix ($n \times 1$).

Network Adjacency Matrix Element

A non-zero element in row $i$, column $j$ represents the number of direct connections/links from node $i$ to node $j$.

Discrete Quantitative Data

Quantitative data consisting of countable distinct values.

Continuous Quantitative Data

Quantitative data consisting of measurable values on a continuous scale.

Displays for Categorical Data

Bar charts and Pie charts.

Stem-and-Leaf Plot Key

A required key explaining the place values (e.g., $2 \text{ | } 3 = 23$).

Distinction Between Column Graph and Histogram

Histogram bars touch each other without spaces between them, except for empty classes or zero frequency.

Ogive Median Reading

On an ogive (cumulative frequency graph), the median is read across from $50\text{\%}$ of the total cumulative frequency on the vertical axis.

Estimated Mean for Grouped Frequency Table

$\bar{x} = \frac{\text{sum}(f \times x)}{\text{sum}(f)}$, where $x$ is the class midpoint and $f$ is the class frequency.

Interquartile Range (IQR)

$\text{IQR} = Q_3 - Q_1$ (Upper Quartile minus Lower Quartile).

Standard Deviation Application

The statistic that best measures spread when the data is symmetrical and has no outliers (often paired with the Mean).

Positively Skewed Distribution

A distribution shape where the mean is pulled higher by the long right tail, resulting in the relationship $\text{Mean} > \text{Median}$.

Outlier Boundaries Formulas

Lower Boundary = $Q_1 - 1.5 \times \text{IQR}$; Upper Boundary = $Q_3 + 1.5 \times \text{IQR}$

Five-Number Summary (Box Plot)

The 5 statistical measures represented on a standard box plot: Minimum, Lower Quartile ($Q_1$), Median ($Q_2$), Upper Quartile ($Q_3$), Maximum.