Premium experiment • trigonometry lab
MATHEMATICS • TRIGONOMETRY
Unit Circle
Connect circular motion to coordinates, sine and cosine waveforms, tangent geometry and exact-angle reasoning while comparing two angles on the same construction.
● LIVE TRIG FIELD45° • quadrant 1
cos θ0.707
sin θ0.707
tan θ1.000
Radians0.785
Locationquadrant 1
Target error0.207
Turn one point into three representations
The same angle is shown as a radius on the circle, Cartesian coordinates (cos θ, sin θ), and aligned sine/cosine traces. That makes periodicity, signs and phase relationships visible at once.
Fast investigations
- Compare 30° and 150° and explain why sine matches while cosine changes sign.
- Rotate through 90° and inspect why tangent becomes undefined.
- Use the target challenge to find the two angles with the same sine value.
- Compare angles separated by 180° and track both coordinate signs.
LEARN WITH THE SIMULATION
Understand Unit Circle by experimenting.
START HERE
The unit circle turns an angle into coordinates: cosine is the horizontal coordinate and sine is the vertical coordinate. This makes trigonometric identities visible rather than purely symbolic.
BUILD THE IDEA
What you should understand first
- Connect angle to (cosθ, sinθ).
- Relate radians to fractions of a full revolution.
- See tangent as sinθ/cosθ and identify where it becomes undefined.
TRY IT YOURSELF
Why does sin²θ + cos²θ always equal 1?
- Move the point to several different angles.
- Record the sine and cosine values.
- Square and add them for each angle.
Read the visual:
Every point lies one unit from the origin, so the Pythagorean theorem gives cos²θ + sin²θ = 1.
WHY IT WORKS 01
Connect angle to (cosθ, sinθ).
The unit circle turns an angle into coordinates: cosine is the horizontal coordinate and sine is the vertical coordinate. This makes trigonometric identities visible rather than purely symbolic. Focus on the relationship between the controlled variable and the measured response rather than treating the visual change as the explanation.
WHY IT WORKS 02
Relate radians to fractions of a full revolution.
Every point lies one unit from the origin, so the Pythagorean theorem gives cos²θ + sin²θ = 1. Use an extreme or boundary setting after the baseline trial to check whether the same reasoning still holds.
DO NOT MIX THESE UP
Common misconception
Degrees and radians are different angle units. Tangent is undefined when cosine is zero. Negative coordinates do not mean negative radius.
PRACTICE CHALLENGE
Why does sin²θ + cos²θ always equal 1?
- Move the point to several different angles.
- Record the sine and cosine values.
- Square and add them for each angle.
Reveal the reasoning
Every point lies one unit from the origin, so the Pythagorean theorem gives cos²θ + sin²θ = 1. Degrees and radians are different angle units. Tangent is undefined when cosine is zero. Negative coordinates do not mean negative radius.
MODEL BOUNDARIES
What this simulation assumes
Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.
Model assumptions
- Angles are represented in an ideal Euclidean plane.
- The circle has exact unit radius and coordinates follow x = cos θ and y = sin θ.
- Displayed decimal values are numerical approximations of exact trigonometric quantities.
Model limitations
- Finite screen resolution and floating-point arithmetic introduce small display rounding.
- The visualization focuses on standard real-valued trigonometry and does not cover complex-angle extensions.
CONNECT IT
Where this matters
- Trigonometric modeling
- Rotations and vectors
- Waves and circular motion
WORDS THAT MATTER
Key vocabulary
- Radian
- Angle subtending an arc equal to the radius.
- Sine
- Vertical coordinate on the unit circle.
- Cosine
- Horizontal coordinate on the unit circle.
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