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Math · Complex Numbers and Parametric Motion

Trig Identities from Unit-Circle Geometry

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Inverse Trig / Angle from RatioRecover angles from ratios while keeping the quadrant signs that made the identity honest.

Key takeaway

Every point on the unit circle satisfies x^2 + y^2 = 1 because the radius is fixed at 1.
Reading that same point as (cos θ, sin θ) gives cos^2 θ + sin^2 θ = 1.
Raw sine or cosine signs can flip across quadrants, but the squared identity survives because the distance from the origin is unchanged.
Complementary first-quadrant angles swap horizontal and vertical projections, so sine and cosine trade roles.

Common misconception

Do not treat trig identities as detached algebra rules. Keep the point, projections, and radius visible so each identity has a geometric reason.

The identities come from the geometry of one point on the unit circle.

Keep this idea moving

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Parametric Curves / Motion from Equations

Keep x(t), y(t), the traced path, and the moving point visible together so shape and traversal stay distinct.

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Derivative as Slope / Local Rate of Change

Slide a point along one curve, tighten a secant into a tangent, and connect local steepness to the derivative graph without leaving the same live bench.

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Reference and support

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Equation snapshotDistance rule on the unit circle · Pythagorean identityShow 3 equations

Read x^2 + y^2 = 1 from the radius first, substitute x = cos θ and y = sin θ second, then use first-quadrant projection swaps for complementary-angle identities.

Distance rule on the unit circle
$x^{2} + y^{2} = 1$
Because the radius stays 1, every point on the circle satisfies this distance relation.
Pythagorean identity
$cos^{2} θ + sin^{2} θ = 1$
Read the same point as $(cos θ, sin θ)$ and the distance relation becomes the identity.
Complementary-angle swap
$sin θ = cos (9 0^{\circ} - θ), cos θ = sin (9 0^{\circ} - θ)$
In the first quadrant, complementary angles swap the horizontal and vertical legs of the same right-triangle geometry, so sine and cosine swap roles.

Why it behaves this way

Explanation

This bench keeps one rotating point, its horizontal and vertical projections, and the squared-projection graph visible together. That way, trig identities come from one piece of geometry instead of feeling like detached symbol rules.

On the unit circle, every point is exactly 1 unit from the origin, so $x^{2} + y^{2} = 1$ is always true. When you read the same point as $(cos θ, sin θ)$ , that becomes $cos^{2} θ + sin^{2} θ = 1$ .

As the point moves through different quadrants, raw sine or cosine can change sign, but the squared sum does not change because the radius does not change. In the first quadrant, complementary angles swap the horizontal and vertical projections, so sine and cosine swap places too.

Key ideas

01Every point on the unit circle satisfies

x^{2} + y^{2} = 1

because the radius is fixed at 1.

02Reading the same point as

(cos θ, sin θ)

turns that distance rule into

cos^{2} θ + sin^{2} θ = 1

03Quadrant changes can flip the sign of sine or cosine, but squaring removes the sign while keeping the same projection lengths in the identity.

04In the first quadrant, complementary angles swap the horizontal and vertical projections, so

sin θ = cos (9 0^{\circ} - θ)

and

cos θ = sin (9 0^{\circ} - θ)

Worked examples

Open examples when you want to see the same idea walked through step by step.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the live point and the squared graph together. First read cosine and sine from the circle, then square them, then compare with the line showing their sum.

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Frozen valuesUsing frozen parameters

For the identity-balance preset, how does the current unit-circle point show that $cos^{2} θ + sin^{2} θ = 1$ ?

θ

Starting angle

53.1 °

1. Read cosine and sine from the point

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 those two values

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

3. Compare with the sum on the graph

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

Live 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.

Quick test

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Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

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 so the identity can be read from one moving point.

Graph summary

One graph shows cosine and sine over time, and the other shows $cos^{2} θ$ , $sin^{2} θ$ , and their sum, which stays at 1 while the point remains on the unit circle.

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Featured setups

See the identity live

Start in Quadrant I with the squared-projection graph open so the point, its projections, and the sum line stay connected.

Test a sign flip

Move the point into Quadrant II and watch cosine change sign while the squared sum still stays at 1.

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Starter track

Step 4 of 6

Complex and Parametric Motion

Trig Identities from Unit-Circle Geometry appears later in this track, so it is cleaner to start from the beginning first.

Previous step: Polar Coordinates / Radius and Angle

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