Coordinate Geometry and Trigonometry

MYP Unit Framework

Key Concept: FORM Related Concepts: Space, Measurement, Justification Global Context: Orientation in Space and Time (How do coordinate systems and trigonometry help us navigate and understand our world?) Statement of Inquiry: Coordinate systems and trigonometric ratios provide tools to describe and analyse spatial relationships.

Inquiry Questions

ATL Skills

• Thinking: Visualise spatial relationships; generalise geometric patterns
• Communication: Use precise mathematical language and notation; construct clear diagrams
• Research: Apply coordinate geometry and trigonometry to real-world measurement problems
• Self-Management: Organise multi-step problem solutions systematically

1. The Cartesian Coordinate System

The Cartesian plane (named after Rene Descartes) is defined by two perpendicular axes:

• x-axis: Horizontal axis
• y-axis: Vertical axis
• Origin (0, 0): The point where axes intersect

Coordinates

Any point is represented as (x, y), where:

• x = horizontal distance from the origin (positive to the right)
• y = vertical distance from the origin (positive upward)

The Four Quadrants

2. Distance and Midpoint

Distance Between Two Points

Given two points (x₁, y₁) and (x₂, y₂):

Distance = sqrt[(x₂ - x₁)² + (y₂ - y₁)²]

This formula is derived from the Pythagorean theorem.

Midpoint of a Line Segment

Midpoint = [(x₁ + x₂)/2, (y₁ + y₂)/2]

The midpoint is the average of the x-coordinates and the average of the y-coordinates.

Applications

• Finding the length of a line segment in a coordinate grid
• Locating the centre point between two locations on a map
• Verifying properties of geometric shapes (e.g., diagonals of a parallelogram bisect each other)

3. Gradient and Equation of a Line (Review and Extension)

Gradient Revisited

Gradient (m) = (y₂ - y₁) / (x₂ - x₁)

Perpendicular Gradients

Two lines with gradients m₁ and m₂ are perpendicular if: m₁ x m₂ = -1

Parallel Lines

Parallel lines have equal gradients: m₁ = m₂

Collinearity

Three or more points are collinear if they lie on the same straight line. This can be verified by checking that the gradient between any two pairs of points is the same.

4. Introduction to Trigonometry

Right-Angled Triangles

A right-angled triangle has one angle of 90 degrees. The sides are:

• Hypotenuse: The longest side, opposite the right angle
• Opposite: The side opposite the angle under consideration
• Adjacent: The side next to the angle under consideration (not the hypotenuse)

The Three Trigonometric Ratios

Sine: sin(theta) = Opposite / Hypotenuse

Cosine: cos(theta) = Adjacent / Hypotenuse

Tangent: tan(theta) = Opposite / Adjacent

Mnemonic: SOH CAH TOA

• Sin = O** / H
• Cos = A / H
• Tan = O / A

Using Trigonometry to Find Missing Sides

Given an angle and one side, use the appropriate ratio to find the missing side.

Example: In a right triangle with angle 30 degrees and hypotenuse 10 cm, find the opposite side.

sin(30) = opposite / 10 → opposite = 10 x sin(30) = 10 x 0.5 = 5 cm

Using Trigonometry to Find Missing Angles

Given two sides, use the inverse trigonometric functions (sin^-1, cos^-1, tan^-1) to find the angle.

Example: If opposite = 4 and adjacent = 3, then tan(theta) = 4/3, so theta = tan^-1(4/3) ≈ 53.1 degrees.

5. Applications of Trigonometry

Angle of Elevation and Depression

• Angle of Elevation: The angle measured upward from the horizontal to an object
• Angle of Depression: The angle measured downward from the horizontal to an object

Real-World Applications

• Surveying: Measuring heights of buildings, trees, or mountains (e.g., finding the height of a building by measuring distance and angle of elevation)
• Navigation: Calculating distances and bearings for ships and aircraft
• Construction: Determining roof pitches, ramp angles, and structural supports
• Astronomy: Calculating distances to celestial objects
• Sports: Analysing the trajectory of a ball or the angle of a golf swing

Bearings

Bearings are directions measured clockwise from north (0 to 360 degrees). Trigonometry is used to resolve bearings into distances and coordinates.

6. Problem-Solving Strategies

Step-by-Step Approach

1. Draw a diagram — always start with a clear sketch
2. Label all known and unknown quantities
3. Identify the appropriate ratio or formula
4. Substitute values and solve
5. Check reasonableness — does the answer make sense?
6. State the answer with appropriate units and precision

Common Mistakes to Avoid

• Using the wrong ratio (confusing opposite and adjacent)
• Calculator in the wrong mode (degrees vs. radians)
• Rounding too early (keep intermediate values exact)
• Forgetting units

Summative Assessment

Task: Mathematical investigation (800-1000 words equivalent) applying coordinate geometry and trigonometry to a real-world problem.

Criteria:

• A: Knowing and Understanding — Select and apply geometric and trigonometric procedures correctly
• B: Investigating Patterns — Identify relationships; generalise and justify
• C: Communicating — Use clear diagrams, notation, and mathematical language
• D: Applying Mathematics in Real-World Contexts — Apply concepts to authentic situations; evaluate the model

Option 1: Survey a real structure (a building, tree, or landmark). Use trigonometry to measure its height. Compare with direct measurement and discuss accuracy.

Option 2: Plan a route on a map using coordinates. Calculate distances between waypoints. Determine bearings for navigation.

Option 3: Design a ramp or roof structure. Use trigonometry to determine angles, lengths, and material needs. Justify your design.

Formative Assessment

• Skills worksheets: distance, midpoint, and gradient calculations
• Trigonometry problem sets: finding missing sides and angles
• Real-world application problems (word problems using diagrams)
• Bearings and navigation exercises
• Practical outdoor measurement activity using clinometers
• Peer assessment of problem-solving approaches

Interdisciplinary Connections

• Physics: Resolving forces into components using trigonometry; projectile motion
• Geography: Using coordinates for mapping; GPS technology; bearings
• Engineering: Structural analysis — forces in trusses and bridges
• Design: Scale drawings; architectural plans; 3D modelling
• Astronomy: Measuring distances to stars using parallax (trigonometric method)

Service as Action

• Accessibility Audit: Survey your school or local area for wheelchair access. Measure ramp angles and determine whether they meet accessibility standards.
• Mapping Project: Create an accurate map of a local park or community space using coordinate geometry. Include key features, distances, and bearings.

IB Learner Profile

• Inquirers: Ask questions about the spatial world and use mathematics to answer them
• Knowledgeable: Understand the relationships between angles, sides, and coordinates
• Thinkers: Apply logical reasoning to solve complex spatial problems
• Communicators: Present solutions clearly using diagrams, notation, and explanation

Self-Test

1. Calculate the distance between (1, 2) and (5, 5).
2. Find the midpoint of the segment joining (-3, 4) and (7, -2).
3. Determine whether points A(1, 3), B(3, 7), and C(5, 11) are collinear.
4. State SOH CAH TOA and explain what each part means.
5. In a right triangle with angle 40 degrees and adjacent side 8 cm, find the opposite side.
6. An observer stands 20 m from a building and measures an angle of elevation of 35 degrees to the top. How tall is the building?
7. What is the difference between angle of elevation and angle of depression?
8. Two lines have gradients of 2/3 and -3/2. Are they perpendicular? Explain.
9. Convert a bearing of 135 degrees into an angle from the x-axis.
10. A ship sails 50 km east, then 30 km north. How far is it from its starting point?

This unit aligns with IB MYP Mathematics guide, developed for Year 4 (Class 9) students.