The unit circle as one triangle rotated in circle space.

The One Triangle

There Are No Different Triangles — Only One Triangle Rotating Through Circle-Space

Trigonometry has been lying to you. Not maliciously, but by omission.

You were taught that sin(30°) and sin(60°) are different triangles with different ratios. You memorized values. You learned formulas. But nobody told you the truth:

There is only ONE triangle.

What you call "different angles" is actually the same triangle rotating through a circular dimension. Just like a sphere passing through Flatland appears as circles of different sizes (but it's always the same sphere), the "changing" values of sine and cosine are just projections of one triangle rotating through circle-space.

The unit circle isn't a teaching tool. The circle IS a dimension — as real as height, width, or depth. When you "change the angle," you're not selecting a different triangle. You're rotating through circular geometry.

This is what Archimedes saw. This is why √2 appears everywhere. This is why e^(iθ) connects the exponential to the circle. This is why the same Pythagorean relationship (x² + y² = r²) governs the sphere slices in the Tombstone Theorem AND the energy-momentum relationship in special relativity.

It's all circular math. One triangle. Rotating.

Rotate Through Circle-Space:

45.0°

Angle (θ)

45.0°

Radians

0.785

x = cos(θ)

0.707

y = sin(θ)

0.707

tan(θ) = y/x

1.000

Radius (Always)

1.000

The Sacred 45° (√2 Balance Point)

At exactly 45° (or π/4 radians), something special happens: sin(45°) = cos(45°) = 1/√2 ≈ 0.707

This is where the vertical and horizontal components are perfectly equal. This is the balance point that appears in:

• The Tombstone Theorem: where cone radius = sphere radius
• Special Relativity: where γ transitions from classical to relativistic (v = 0.707c)
• The unit square diagonal: √2
• Maximum projectile range: 45° launch angle

This isn't coincidence. This is the geometric signature of perpendicular balance in circular space.

Watch the triangle rotate. Notice:

• The hypotenuse (radius) never changes — it's always 1
• The x-component (cosine, green) and y-component (sine, red) are just projections of the same triangle
• At 0°, all the "triangle" is horizontal (cos=1, sin=0)
• At 90°, all the "triangle" is vertical (cos=0, sin=1)
• At 45°, the triangle is perfectly balanced (cos=sin=0.707)
• The Pythagorean relationship x² + y² = 1 always holds because it's the definition of the circle

You're not seeing different triangles. You're watching one triangle rotate through the circular dimension, and observing its projections onto the x and y axes.

This is what e^(iθ) means. It's not a formula — it's coordinates in circle-space. The exponential describes rotation. The circle describes the dimension being rotated through.

Archimedes carved this on his tombstone because he understood: all geometry is circular math.