Turn an angle into two coordinates
Why can sine or cosine be negative when a triangle side length cannot?
Entering midway? A full turn is 360 degrees or 2π radians. A radius is the distance from the origin to the circle.
Start with a concrete question
A rotating radius defines horizontal and vertical coordinates. Inspect signed coordinates in every quadrant before using right-triangle ratios.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict which coordinate changes sign when the angle moves from 30° to 150°.
Change one thing. Inspect what happens.
- At 30°, compare the radius and the vertical leg.
- Compare 30°, 150° and −30°; explain signs using position.
- Set 90° and inspect why tangent is undefined while sine and cosine remain defined.
- Change radius without changing angle, then set radius zero and distinguish defined angle functions from undefined coordinate/radius ratios.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Radius, circle and coordinate legs — equal coordinate units
Horizontal: Horizontal coordinate (m). Vertical: Vertical coordinate (m).
- 1. Circle (solid line)
- 2. Radius (dashed line)
- 3. Coordinate legs (dashed line)
Axes use the displayed ranges.
Read plotted values as a table
| Series | Horizontal coordinate (m) | Vertical coordinate (m) |
|---|---|---|
| Circle | 2 | 0 |
| Circle | 1.992 | 0.1743 |
| Circle | 1.97 | 0.3473 |
| Circle | 1.932 | 0.5176 |
| Circle | 1.879 | 0.684 |
| Circle | 1.813 | 0.8452 |
| Circle | 1.732 | 1 |
| Circle | 1.638 | 1.147 |
| Circle | 1.532 | 1.286 |
| Circle | 1.414 | 1.414 |
| Circle | 1.286 | 1.532 |
| Circle | 1.147 | 1.638 |
| Circle | 1 | 1.732 |
| Circle | 0.8452 | 1.813 |
| Circle | 0.684 | 1.879 |
| Circle | 0.5176 | 1.932 |
| Circle | 0.3473 | 1.97 |
| Circle | 0.1743 | 1.992 |
| Circle | 1.22e-16 | 2 |
| Circle | -0.1743 | 1.992 |
| Circle | -0.3473 | 1.97 |
| Circle | -0.5176 | 1.932 |
| Circle | -0.684 | 1.879 |
| Circle | -0.8452 | 1.813 |
| Circle | -1 | 1.732 |
| Circle | -1.147 | 1.638 |
| Circle | -1.286 | 1.532 |
| Circle | -1.414 | 1.414 |
| Circle | -1.532 | 1.286 |
| Circle | -1.638 | 1.147 |
| Circle | -1.732 | 1 |
| Circle | -1.813 | 0.8452 |
| Circle | -1.879 | 0.684 |
| Circle | -1.932 | 0.5176 |
| Circle | -1.97 | 0.3473 |
| Circle | -1.992 | 0.1743 |
| Circle | -2 | 2.45e-16 |
| Circle | -1.992 | -0.1743 |
| Circle | -1.97 | -0.3473 |
| Circle | -1.932 | -0.5176 |
| Circle | -1.879 | -0.684 |
| Circle | -1.813 | -0.8452 |
| Circle | -1.732 | -1 |
| Circle | -1.638 | -1.147 |
| Circle | -1.532 | -1.286 |
| Circle | -1.414 | -1.414 |
| Circle | -1.286 | -1.532 |
| Circle | -1.147 | -1.638 |
| Circle | -1 | -1.732 |
| Circle | -0.8452 | -1.813 |
| Circle | -0.684 | -1.879 |
| Circle | -0.5176 | -1.932 |
| Circle | -0.3473 | -1.97 |
| Circle | -0.1743 | -1.992 |
| Circle | -3.67e-16 | -2 |
| Circle | 0.1743 | -1.992 |
| Circle | 0.3473 | -1.97 |
| Circle | 0.5176 | -1.932 |
| Circle | 0.684 | -1.879 |
| Circle | 0.8452 | -1.813 |
| Circle | 1 | -1.732 |
| Circle | 1.147 | -1.638 |
| Circle | 1.286 | -1.532 |
| Circle | 1.414 | -1.414 |
| Circle | 1.532 | -1.286 |
| Circle | 1.638 | -1.147 |
| Circle | 1.732 | -1 |
| Circle | 1.813 | -0.8452 |
| Circle | 1.879 | -0.684 |
| Circle | 1.932 | -0.5176 |
| Circle | 1.97 | -0.3473 |
| Circle | 1.992 | -0.1743 |
| Circle | 2 | -4.90e-16 |
| Radius | 0 | 0 |
| Radius | 1.732 | 1 |
| Coordinate legs | 0 | 0 |
| Coordinate legs | 1.732 | 0 |
| Coordinate legs | 1.732 | 1 |
Follow the calculation
- Radians = degrees × π/180 = 30 × π/180 = 0.52360.
- cos θ is horizontal coordinate / radius; sin θ is vertical coordinate / radius on a nonzero circle.
- x = 2 × 0.86603 = 1.73205 m; y = 2 × 0.50000 = 1.00000 m.
- cos² θ + sin² θ = 1.00000.
- tan θ = sin θ/cos θ = 0.57735; this is a signed ratio, not a length.
Sine and cosine are signed coordinate ratios. In the first quadrant they match opposite/hypotenuse and adjacent/hypotenuse in a right triangle.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Start with the meaning
A full turn is 360 degrees or 2π radians. A radius is the distance from the origin to the circle.
A rotating radius defines horizontal and vertical coordinates. Inspect signed coordinates in every quadrant before using right-triangle ratios.
A worked explanation
On the unit circle, cosine is the horizontal coordinate and sine the vertical coordinate. A radius r scales both coordinates.
At 30°=π/6 radians, sin θ=1/2 and cos θ=√3/2. With radius two, the endpoint is (√3,1). In the first quadrant these are the adjacent and opposite legs divided by the hypotenuse.
Beyond the first quadrant, signs encode direction. The identity sin²θ+cos²θ=1 follows from the unit-circle distance equation.
Symbols and relationship
Radians=degrees×π/180; x=r cos θ; y=r sin θ. tan θ=sin θ/cos θ only when cos θ≠0.
Use the graph and trace together
The graph and calculations use your current inputs. Read axis labels and units before comparing positions or slopes. Change one input, predict the effect, and use the numerical trace to verify the plotted behavior. A displayed curve is an illustration; the formula and its assumptions determine which conclusions are justified.
Scope and edge cases
- Angles are measured counterclockwise from the positive horizontal axis.
- Zero radius collapses all endpoints; the coordinate direction cannot be recovered from that point.
Try a fresh case
A radius of 6 m makes an angle of 30° with the positive horizontal axis. What is its vertical coordinate?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Draw coordinate triangles in all four quadrants and label signs.
College investigation
Derive angle-addition formulas using rotations, then compare with the existing geometry support lab.
Workplace investigation
Write angle-unit and axis-convention checks for a coordinate interface.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.