Maths Applications Test Revision

Test Criteria

Topic 2.2: Applications of trigonometry 

2.2.2     determine the area of a triangle, given two sides and an included angle by using the rule A=1/2 absinC , or given three sides by using Heron’s rule, and solve related practical problems

2.2.3     solve problems involving non-right-angled triangles using the sine rule (acute triangles only when determining the size of an angle) and the cosine rule

2.2.4     solve practical problems involving right-angled and non-right-angled triangles, including problems involving angles of elevation and depression and the use of bearings in navigation

Topic 2.3: Linear equations and their graph

2.3.1     identify and solve linear equations (with the aid of technology where complicated manipulations are required)

2.3.2     develop a linear formula from a word description and solve the resulting equation

Straight-line graphs and their applications

2.3.3     construct straight-line graphs both with and without the aid of technology

2.3.4     determine the slope and intercepts of a straight-line graph from both its equation and its plot

2.3.5     construct and analyse a straight-line graph to model a given linear relationship; for example, modelling the cost of filling a fuel tank of a car against the number of litres of petrol required.

2.3.6     interpret, in context, the slope and intercept of a straight-line graph used to model and analyse a practical situation

applications

2.3.7     solve a pair of simultaneous linear equations graphically or algebraically, using technology when appropriate

2.3.8     solve practical problems that involve determining the point of intersection of two straight-line graphs; for example, determining the break-even point where cost and revenue are represented by linear equations

Piece-wise linear graphs and step graphs

2.3.9     sketch piece-wise linear graphs and step graphs, using technology when appropriate

2.3.10   interpret piece-wise linear and step graphs used to model practical situations; for example, the tax paid as income increases, the change in the level of water in a tank over time when water is drawn off at different intervals and for different periods of time, the charging scheme for sending parcels of different weights through the post

📘 Topic 2.2 – Applications of Trigonometry

2.2.2 Area of a Triangle

1⃣ Formula (Two sides and included angle)

When you know two sides and the included angle (the angle between those two sides):

Where:

• aaa and bbb are sides of the triangle

• CCC is the included angle between them

• AAA is the area

2⃣ Heron’s Formula (Three sides known)

When you know all three sides, use Heron’s formula.

is the semi-perimeter.

Example:

2.2.3 Solving Non-Right-Angled Triangles

Sine Rule

Used when you know:

Cosine Rule

Used when you know:

• Two sides and the included angle (SAS) to find the third side, or

• All three sides (SSS) to find an angle.

2.2.4 Practical Trig Problems (Elevation, Depression, Bearings)

Angles of Elevation and Depression

• Elevation: angle upwards from the horizontal

• Depression: angle downwards from the horizontal
  → Both measured from a horizontal line

Use sine, cosine, or tangent as in right-angled triangles, or sine/cosine rules for non-right-angled ones.

Example (Right-angled):

Bearings

• Bearings are measured clockwise from North

• Always written as three digits (e.g. 045°, 120°, 270°)

Example (Navigation):
Ship A is 20 km north of Ship B. Ship A sails 15 km on a bearing of 120°. Find the distance between the ships.

→ Draw diagram, apply cosine rule between points.

📘 Topic 2.3 – Linear Equations and their Graphs

2.3.1 Identify and Solve Linear Equations

A linear equation is one where the variable has power 1.

Example:

Use technology (e.g. calculator or CAS) if equations are complex (fractions, decimals).

2.3.2 Develop Linear Formula from Word Problem

Example:

2.3.3 Construct Straight-Line Graphs

2.3.4 Determine Slope and Intercepts

From equation y=mx+cy = mx + cy=mx+c:

• Slope = mmm

• y-intercept = ccc

• To find x-intercept → let y=0y = 0y=0

Example:

2.3.5 & 2.3.6 Modelling Linear Relationships

Use straight-line graphs to model real-life scenarios.

Example:

2.3.7 Solve Simultaneous Equations

Two linear equations with two unknowns — find the point of intersection.

Example:

2.3.8 Practical Applications (Break-even point)

Example:

2.3.9 & 2.3.10 Piece-wise and Step Graphs

Used when the relationship changes in different ranges.

Piece-wise Linear Graph:

Graph with different line equations for different intervals.

Step Graph:

Graph made of flat (horizontal) sections—used for things like postage or tax brackets.

Example:

A postage company charges:

• $5 for parcels up to 2 kg

• $8 for parcels 2–5 kg

• $10 for parcels over 5 kg

→ Graph is a step function: horizontal segments that “jump” at 2 kg and 5 kg.

Interpretation Example:

At 4 kg, cost = $8
The jump from $8 → $10 at 5 kg represents an increased rate for heavier parcels.

✅ Summary Table