How the Unit Circle Works: Sin, Cos and Tan

Quick Answer

The unit circle is a circle of radius 1 centered at the origin, and the point at angle θ has coordinates exactly (cos θ, sin θ). That single fact defines sine and cosine for every angle. For example, at 45° the point sits at (0.707, 0.707), so sin and cos are both √2/2. Tangent is the ratio tan θ = sin θ / cos θ. Open the explorer, drag the point, and read sin cos tan values live as the reference triangle follows your angle.

How the Unit Circle Works

Draw a circle of radius 1 and put an angle θ at the center, measured counter-clockwise from the positive x-axis. Some textbooks simply call it the trig circle. The radius arm meets the circle at one point, and that point's coordinates are the whole story: x = cos θ and y = sin θ. Drop a vertical line to the x-axis and you get the reference triangle, whose horizontal leg is the cosine and vertical leg is the sine. Because the hypotenuse is 1, the Pythagorean theorem becomes the identity sin² θ + cos² θ = 1, which is why sin and cos can never leave the range -1 to 1.

• The horizontal coordinate of the point is cos θ
• The vertical coordinate of the point is sin θ
• tan θ = sin θ / cos θ, the slope of the radius arm

Radians and Degrees

Degrees chop a full turn into 360 equal parts because the Babylonians liked the number; radians measure the angle by the arc it cuts on the unit circle. A full turn is an arc of 2π (the circumference), a half turn is π, and a right angle is π/2. For example, 60° is π/3 radians, an arc exactly one third of π long. To convert radians and degrees by hand, multiply radians by 180/π or degrees by π/180. However, the explorer shows radians and degrees side by side while you drag, which is the fastest way to build the mental conversion table without formulas.

Special Angles and Reference Angles

Three acute angles carry exact values worth memorizing: at 30° the values are sin θ = 1/2 and cos θ = √3/2, at 45° both equal √2/2, and at 60° they swap to sin θ = √3/2 and cos θ = 1/2. Every other angle on the classic chart reduces to one of these special angles through its reference angle, the acute angle between the radius arm and the x-axis. For example, 150° has reference angle 30°, so its sine is still 1/2 while its cosine flips sign to -√3/2 because the point is now left of the y-axis. Quadrants only change signs, never magnitudes.

Where the Sine Wave Comes From

Set a rotation speed and watch the right side of the canvas: as the point circles at a steady rate, its height traced against the angle draws a perfect sine wave. This is the deep link between circular motion and oscillation. A wheel turning, an alternating current, and a speaker cone all move in circles or cycles, and therefore their graphs are sine waves. Programmers lean on the same picture when rotating sprites in games or graphics: every rotation is a walk around this circle. The dashed line from the point to the wave shows the projection happening in real time, and you can also treat the page as a unit circle calculator: pause, type or drag an exact angle, and read the values.

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