Interactive guide
Sine and cosine aren't formulas. They're coordinates.
Walk a point around a circle of radius 1, centered on the origin. At every angle, that point sits at some (x, y). Cosine is just its x-coordinate. Sine is just its y-coordinate. That's the entire definition, the triangle-and-SOHCAHTOA version from school is a special case of this, not the other way around.
Why radians, not just degrees? A degree is an arbitrary slice, a full turn cut into 360 pieces for historical reasons. A radian is the angle where the arc length walked around the circle equals the radius itself. Because the circle's radius is exactly 1 here, an angle in radians is the distance traveled along the rim. That's why calculus almost always switches to radians: derivatives of sine and cosine only come out clean (sin′ = cos, not some messy constant times cos) when angle is measured this way.
Drag the point. Read off the coordinates.
Below, dragging the point around the circle changes the angle θ (theta), measured counter-clockwise from the positive x-axis. The dashed legs show exactly what cosine and sine mean at that angle: cosθ is the horizontal leg, sinθ is the vertical one.
Underneath, the same angle drives two live charts. As θ sweeps around the circle over and over, sinθ and cosθ trace out the wave shapes you've seen before, because that's literally all a sine wave is: one coordinate of a point going around a circle, unrolled against the angle.
Tangent gets its own chart, because it behaves nothing like the other two: tanθ = sinθ / cosθ, and whenever cosθ hits exactly 0 (at 90° and 270°), that division blows up. Watch the tangent curve shoot off the top or bottom of its chart at exactly those two angles, every single lap.
The three ratios, defined once
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1
cosθ = x
The point's horizontal position on a radius-1 circle. Always between -1 and 1.
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2
sinθ = y
The point's vertical position. Also always between -1 and 1.
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3
tanθ = sinθ / cosθ
The slope of the radius line itself. Undefined wherever cosθ = 0, since that's division by zero.
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4
Every 360° (2π rad), it repeats
Walking a full lap brings the point back to where it started, so all three ratios are periodic.
Drag the point around the circle.
adjacent = |cosθ| opposite = |sinθ| hypotenuse = 1 (drag me)
Sine & cosine, unrolled against the angle
cosθ sinθ
Tangent, its own chart (asymptotes dashed)
tanθ asymptote
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- θ
- 0°
- Quadrant
- —
- Reference angle
- 0°
Try dragging past 90° and 270°. Nothing dramatic happens to the circle, the point just keeps sliding along, but watch the tanθ readout: it doesn't pass through some large number on its way, it jumps straight from a huge positive value to a huge negative one, because cosθ (the adjacent side, and tanθ's denominator) crossed zero and flipped sign right at those two spots. Notice too that the cosine curve above is just the sine curve, shifted a quarter turn (90°, or π/2 rad) to the left, cosθ = sin(θ + 90°).
This same cosθ, sinθ pair is exactly what a rotation matrix is built from: [[cosθ, -sinθ], [sinθ, cosθ]] rotates any vector by θ. It's why the Rotation preset on the Eigenvalues & Eigenvectors page has no real eigenvectors at all: it turns every direction by the same angle, exactly what dragging the point all the way around this circle just did.
The jargons
The unit circle is a circle of radius 1 centered at the origin. Everything below is just a name for some fact about a point walking around it.
The angle where arc length traveled equals the radius. On the unit circle, an angle's radian measure is literally the distance walked along the rim.
The y-coordinate of the point at angle θ on the unit circle.
The x-coordinate of that same point.
sinθ / cosθ, the slope of the radius line. Undefined wherever cosθ = 0.
Which quarter of the plane the point sits in (I through IV), which fixes the sign of sin, cos, and tan without knowing the exact angle.
The acute angle between the radius line and the nearest x-axis, the number you'd look up in a table before fixing its sign for the quadrant.
A line a curve approaches but never crosses. Tangent has one at every angle where cosine is 0.
Sine and cosine repeat every 360° (2π rad); tangent repeats twice as often, every 180° (π rad).