Year 9 Mathematics Comprehensive Exam Reference Guide

The Hypotenuse: In a right-angled triangle, the hypotenuse is defined as the longest side. It is always located directly opposite the $90^{\circ}$ (right) angle.
Opposite and Adjacent Sides: These sides are defined relative to a specific angle, often denoted as $\theta$ . - The Opposite side is the side directly across from the angle $\theta$ . - The Adjacent side is the side next to the angle $\theta$ that is not the hypotenuse.
Standard Trigonometric Ratios (SOH CAH TOA): - SOH: Sine is defined as the ratio of the Opposite side to the Hypotenuse. - $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ - CAH: Cosine is defined as the ratio of the Adjacent side to the Hypotenuse. - $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ - TOA: Tangent is defined as the ratio of the Opposite side to the Adjacent side. - $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

Standard Step-by-Step Workflow: - Step 1: Label the sides of the triangle as Opposite ( $O$ ), Adjacent ( $A$ ), and Hypotenuse ( $H$ ) based on their position relative to the given or unknown angle. - Step 2: Identify which sides are involved in the problem (e.g., Which side do I have the length for? Which side do I need to find?). - Step 3: Select the appropriate ratio from SOH CAH TOA based on the identified sides. - Step 4: Set up the equation and solve carefully.
Finding a Missing Side: - Use the standard ratios: $\sin(\theta) = \frac{O}{H}$ , $\cos(\theta) = \frac{A}{H}$ , or $\tan(\theta) = \frac{O}{A}$ .
Finding a Missing Angle: - To determine the value of an unknown angle, the inverse trigonometric functions must be utilized: - $\sin^{-1}$ - $\cos^{-1}$ - $\tan^{-1}$

Required Mode: Ensure the calculator is set to DEG (Degree) mode for all trigonometric calculations.
Entering Degrees and Minutes: To input an angle such as $23^{\circ}18'$ (23 degrees and 18 minutes), use the degrees, minutes, and seconds key, typically labeled as ${^\circ\prime\prime}$ . - Typing Sequence: $23 \rightarrow {^\circ\prime\prime} \rightarrow 18 \rightarrow {^\circ\prime\prime}$
Calculation Example: If a calculation requires finding the sine of $23^{\circ}18'$ and multiplying it by a factor of $57$ , the syntax would be: - $\sin(23^{\circ}18') \times 57$

Definition: Pythagoras' Theorem describes the relationship between the three sides of a right-angled triangle.
Formula: $a^2 + b^2 = c^2$
Variable Identification: In this formula, $c$ represents the length of the hypotenuse (the longest side). The variables $a$ and $b$ represent the lengths of the two shorter sides (legs).

Expanding Expressions: This involves removing parentheses by multiplying the term outside the brackets by every term inside the brackets. - Example: $3(x + 2) = 3x + 6$
Factorising Expressions: This is the inverse process of expansion, where a common factor is identified and moved outside of a bracketed expression. - Example: $6x + 12 = 6(x + 2)$

Adding Fractions: To add two or more fractions together, they must possess the same denominators.
Multiplying Fractions: Unlike addition, common denominators are not required. The numerators are multiplied together, and the denominators are multiplied together. - Rule: $\frac{\text{top} \times \text{top}}{\text{bottom} \times \text{bottom}}$

Calculating a Percentage of a Quantity: To find a specific percentage of a total value, convert the percentage to a decimal and multiply by the total. - Example: To find $20\%$ of $50$ , calculates as $0.2 \times 50$ .
Goods and Services Tax (GST): - To add GST to a price (assuming a standard $10\%$ increase), multiply the original cost by a factor of $1.1$ . - Formula: $\text{Price incl. GST} = \text{Original Price} \times 1.1$
Profit Calculation: Profit is determined by the difference between the final selling price and the initial cost price. - Formula: $\text{Profit} = \text{Selling Price} - \text{Cost Price}$