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Discover trigonometry · 45 minutes

At 30° on the unit circle, y = 0.5. Predict x and y at 120° before turning the arm.

Investigation and discussion	Time
Arrival	3 minutes
1 · Triangles and ratios	5 minutes
2A · Circle and projections	5 minutes
2B · Circle to wave	5 minutes
3 · Beam and wall	5 minutes
4 · Radians	4 minutes
5 · Lighthouse height	5 minutes
Paired comparison	8 minutes
Exit explanation	5 minutes
Total	45 minutes

Collect explanations

The radius doubles. Which values change, and why?
The angle changes from 30° to 120°. Which signs change?
Why does a 120° forward ray miss the wall although tangent is defined?
Solve sin θ = 0.5 on [0°, 720°). Why is the principal value insufficient?
What assumptions support the survey result? How does uncertainty affect it?

Preparation and access

No account or sign-in is required. Choose Diagram for exact plots or lower data use. Every experiment has numeric controls and value tables. Less motion retains spatial viewing with deliberate steps. Share experiments with links; student reflections are never included in the link.

Prepared examples and investigation worksheets

Each link opens a prepared example in diagram view. Choose 3D inside the lab when it suits your class. Open a station brief to see its goal, assumptions and tasks.

Triangles and ratios

Learning goal: Explain why side ratios remain unchanged when a triangle grows.
Prior knowledge: Recognise a right angle, side lengths and simple ratios.
Model assumptions: The reference angle lies strictly between 0° and 90° and the radius is positive.
Edge case: At 0° and 90° the shape is degenerate; another-quadrant angle is outside this acute triangle example.
Transfer: Set r = 1.5 and θ = 30°. Predict the opposite side and explain your answer using a ratio.

Open prepared example

What changes when a triangle grows?

1 · Predict. Predict the opposite side and opposite / hypotenuse ratio when the radius doubles.

2 · Experiment. Change one quantity at a time. Record the settings and results, then compare them with your prediction.

3 · Explain. Explain the relationship using a drawing, words or a formula.

4 · Changed case. Set r = 1.5 and θ = 30°. Predict the opposite side and explain your answer using a ratio.

2A · Circle and projections

Learning goal: Connect sin θ and cos θ with signed coordinates and their ratios to radius.
Prior knowledge: Read coordinates and distinguish horizontal from vertical directions.
Model assumptions: The angle is counterclockwise from the positive x-axis; lengths use model units.
Edge case: At 120°, x is negative but the side length |x| is positive.
Transfer: Compare 150° and 330°. Explain the coordinate signs and what changes when r doubles.

Open prepared example

What do the point's coordinates tell us about the angle?

1 · Predict. At 30° on the unit circle, y = 0.5. Predict x and y at 120° before turning the arm.

2 · Experiment. Change one quantity at a time. Record the settings and results, then compare them with your prediction.

3 · Explain. Explain the relationship using a drawing, words or a formula.

4 · Changed case. Compare 150° and 330°. Explain the coordinate signs and what changes when r doubles.

2B · Circle to wave

Learning goal: Distinguish angle and time graphs and connect constant angular speed with a repeating wave.
Prior knowledge: Connect vertical coordinates with sin θ and read graph axes.
Model assumptions: The time model uses constant angular speed and a starting phase; manually chosen θ is independent.
Edge case: Arbitrary handle movement is not a time experiment at constant angular speed.
Transfer: Double ω on the time graph. Predict the period and explain why the amplitude stays unchanged.

Open prepared example

How does height repeat during rotation?

1 · Predict. Predict the point's height at 0°, 90°, 180°, 270° and 360°. Imagine how height changes between them.

2 · Experiment. Change one quantity at a time. Record the settings and results, then compare them with your prediction.

3 · Explain. Explain the relationship using a drawing, words or a formula.

4 · Changed case. Double ω on the time graph. Predict the period and explain why the amplitude stays unchanged.

Beam and wall

Learning goal: Connect tan θ with rise/run and distinguish a line slope from a forward wall hit.
Prior knowledge: Know sin θ, cos θ and division.
Model assumptions: The wall lies to the right of the origin at distance d > 0.
Edge case: At 89° the finite height may be outside the view; at 90° tangent is undefined.
Transfer: Compare 30° and 210°. Explain their equal slopes and different forward-ray results.

Open prepared example

When does the beam hit the wall?

1 · Predict. The wall is to the right of the origin. Predict what happens at 90° and 120°.

2 · Experiment. Change one quantity at a time. Record the settings and results, then compare them with your prediction.

3 · Explain. Explain the relationship using a drawing, words or a formula.

4 · Changed case. Compare 30° and 210°. Explain their equal slopes and different forward-ray results.

An angle wrapped around a circle

Learning goal: Explain radians through s/r and distinguish directed rotation from circle position.
Prior knowledge: Recognise radius, arc and circumference.
Model assumptions: In s = rθ, θ uses radians and s is directed arc displacement from the reference position.
Edge case: 60° and 420° share an endpoint but have different net rotations; displacement is not total travel after reversals.
Transfer: Double radius at a fixed angle. Explain which lengths double and which ratio stays unchanged.

Open prepared example

How many radius lengths fit along the arc?

1 · Predict. An arc as long as the radius subtends one radian. What happens to the angle when both lengths double?

2 · Experiment. Change one quantity at a time. Record the settings and results, then compare them with your prediction.

3 · Explain. Explain the relationship using a drawing, words or a formula.

4 · Changed case. Double radius at a fixed angle. Explain which lengths double and which ratio stays unchanged.

Surveying the coast

Learning goal: Infer height from horizontal distance and elevation and explain assumptions and uncertainty.
Prior knowledge: Know tangent as opposite/adjacent.
Model assumptions: The fictional tower is vertical and 31.6 m high; ground is level, d is measured in metres and instrument height must be added.
Edge case: 90° or an uncertainty interval reaching 90° has no finite upper height bound; SSA data may admit two triangles.
Transfer: Move the observation point from 40 m to 20 m. Which angle still recovers 31.6 m, and why? Then compare two SSA solutions.

Open prepared example

How can you find a height without climbing?

1 · Predict. We observe from level ground in this fictional example. What measurements do you need besides the elevation angle?

2 · Experiment. Change one quantity at a time. Record the settings and results, then compare them with your prediction.

3 · Explain. Explain the relationship using a drawing, words or a formula.

4 · Changed case. Move the observation point from 40 m to 20 m. Which angle still recovers 31.6 m, and why? Then compare two SSA solutions.

The equation that stays true

Learning goal: Justify sin² θ + cos² θ = 1 through Pythagoras and distinguish it from x² + y².
Prior knowledge: Understand square areas, coordinates and Pythagoras.
Model assumptions: r > 0 allows division of x² + y² = r² by r²; areas remain nonnegative when coordinates are negative.
Edge case: At r = 2, x² + y² = 4 but cos² θ + sin² θ = 1; sin θ + cos θ is a different expression.
Transfer: Test 120° at r = 2 and justify the result. Add 45° and compare with the sum-angle identities.

Open prepared example

Why is sin² θ + cos² θ always 1?

1 · Predict. Turn the arm. Predict the sum of the square areas on the horizontal and vertical sides.

2 · Experiment. Change one quantity at a time. Record the settings and results, then compare them with your prediction.

3 · Explain. Explain the relationship using a drawing, words or a formula.

4 · Changed case. Test 120° at r = 2 and justify the result. Add 45° and compare with the sum-angle identities.

Wave workshop

Learning goal: Relate amplitude, period, phase and centre to separate wave controls.
Prior knowledge: Read a time axis and recognise sine as a repeating function.
Model assumptions: y(t) = c + A sin(ωt + φ) is an ideal model; φ uses radians in the formula and ω uses radians per second.
Edge case: A = 0 or ω = 0 gives a constant with no unique fundamental period; negative ω reverses motion.
Transfer: Find amplitude 2, period 4 s and centre −1. Explain each parameter change before trying it.

Open prepared example

Which control changes which wave feature?

1 · Predict. In y(t) = c + A sin(ωt + φ), A controls amplitude. Which control changes the period?

2 · Experiment. Change one quantity at a time. Record the settings and results, then compare them with your prediction.

3 · Explain. Explain the relationship using a drawing, words or a formula.

4 · Changed case. Find amplitude 2, period 4 s and centre −1. Explain each parameter change before trying it.

One value, several angles

Learning goal: Distinguish a principal inverse value from all equation solutions in a stated interval.
Prior knowledge: Know circle coordinates, sine and repetition after a full turn.
Model assumptions: q is dimensionless; the solution window is half-open and excludes its right endpoint.
Edge case: At q = ±1 the two families coincide; |q| > 1 has no real sine or cosine solution.
Transfer: Solve cos θ = 0.5 over two turns. Then solve 1 + 2 sin(πt/2) = 2 on [0,4) s and justify the interval list.

Open prepared example

Why does a calculator give just one angle?

1 · Predict. Find every angle in [0°, 360°) satisfying sin θ = 0.5. Is arcsin(0.5) a complete answer?

2 · Experiment. Change one quantity at a time. Record the settings and results, then compare them with your prediction.

3 · Explain. Explain the relationship using a drawing, words or a formula.

4 · Changed case. Solve cos θ = 0.5 over two turns. Then solve 1 + 2 sin(πt/2) = 2 on [0,4) s and justify the interval list.

Direction and rotation

Learning goal: Connect direction and magnitude with components and explain length preservation under rotation.
Prior knowledge: Read coordinates and know sin θ and cos θ.
Model assumptions: The vector and rotation are planar; translating the origin changes endpoint coordinates without changing components.
Edge case: The zero vector has no unique direction; a magnitude above 3 remains a valid local vector although transfer to the shared view is unavailable.
Transfer: Move the origin to (1,−1), enter components (−1,1) and rotate by 45°. Explain direction, endpoint and preserved magnitude.

Open prepared example

What stays unchanged when a vector rotates?

1 · Predict. A unit vector at 30° has components (0.866, 0.5). What changes after a 45° rotation?

2 · Experiment. Change one quantity at a time. Record the settings and results, then compare them with your prediction.

3 · Explain. Explain the relationship using a drawing, words or a formula.

4 · Changed case. Move the origin to (1,−1), enter components (−1,1) and rotate by 45°. Explain direction, endpoint and preserved magnitude.

Source and curriculum connections

Primary-school connections include similarity, ratios, circles and Pythagoras; the reviewed chapter 25 does not explicitly name trigonometry. Secondary-school connections include trigonometric functions at competency level 2 and trigonometric equations at level 3. A competency level is not a school year. These are proposed formative tasks, not validated assessment instruments.

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