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Concept module

Trig Identities from Unit-Circle Geometry

Keep one rotating point and its projections visible so the core trig identities stay tied to geometry instead of detached symbol rules.

Interactive lab

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Featured setups

Identity balance

Open a first-quadrant point with the squared-projection graph already in view.

Quadrant II balance

Move the point left of the y-axis and watch the squared sum stay fixed anyway.

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Stable links

Starter track

Step 4 of 60 / 6 complete

Complex and Parametric Motion

Earlier steps still set up Trig Identities from Unit-Circle Geometry.

1. Complex Numbers on the Plane2. Unit Circle / Sine and Cosine from Rotation3. Polar Coordinates / Radius and Angle4. Trig Identities from Unit-Circle Geometry+2 more steps

Previous step: Polar Coordinates / Radius and Angle.

Start from track beginning

Why it behaves this way

Explanation

Trig identities land more honestly when they stay on the same unit-circle bench as the rotating point. This page keeps the point, its projections, and a squared-projection graph visible together so the identities come from geometry instead of looking like detached symbol tricks.

On the unit circle the radius is always 1, so every live point satisfies x^2 + y^2 = 1. Once the same point is read as (cos theta, sin theta), the Pythagorean identity is forced by the picture. Complementary-angle swaps are geometric too: in the first quadrant, swapping the horizontal and vertical shadows swaps cosine and sine.

Key ideas

01Because the unit-circle radius is 1, every live point satisfies x^2 + y^2 = 1.
02Replacing x with cos theta and y with sin theta gives cos^2 theta + sin^2 theta = 1.
03Raw sine and cosine signs can change across quadrants, but the squared sum stays fixed because the distance from the origin does not change.
04Complementary angles swap the horizontal and vertical projections in the first quadrant, which is why sine and cosine trade places.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough
Use the identity-balance preset so the point and the squared graph stay aligned.

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Frozen valuesUsing frozen parameters

For the identity-balance preset, what does the unit-circle point say about the core identity?

Starting angle

53.1 °

1. Read the live projections

At about 53.1 deg, the unit-circle point is close to (0.60, 0.80), so cos theta is about 0.60 and sin theta is about 0.80.

2. Square the same two projections

That gives cos^2 theta \approx 0.36 and sin^2 theta \approx 0.64.

3. Add the squared pieces

The squared-projection graph shows the same result numerically: 0.36 + 0.64 = 1.00.

Identity check

\cos^2\theta + \sin^2\theta = 1
The identity is not a separate algebra trick. It is the distance formula for one point that never leaves the unit circle.

Common misconception

Trig identities are mostly symbolic algebra, so the unit-circle picture is optional once the formulas are memorized.

The identities come from the same geometry as the live projections.

If the point stays on the unit circle, the squared projections must still add to 1 even when the raw signs change.

Mini challenge

Move the point from Quadrant I into Quadrant II without changing its distance from the origin. What has to stay true about the squared projections?

Make a prediction before you reveal the next step.

Decide whether the sign change should affect the squared sum or only the raw cosine sign.

Check your reasoning against the live bench.

The squared sum still has to stay equal to 1.
Crossing into Quadrant II flips the raw cosine sign, but the point is still on the same unit circle, so the distance formula does not change.

Quick test

Misconception check

Question 1 of 2

Answer from the live point and the squared graph.

What is the cleanest geometric source of cos^2 theta + sin^2 theta = 1 on this bench?

Use the live bench to test the result before moving on.

Accessibility

The simulation shows a unit circle with one rotating point, horizontal and vertical projection guides, an angle marker, a quadrant sign map, and graphs that compare the raw projections with their squared values.

Graph summary

One graph shows cosine and sine changing over time, and a second graph shows cosine squared, sine squared, and their sum staying fixed at 1.