How this instrument works
Cosine is defined from a circle of radius 1 centered on the origin: sweep an angle θ counterclockwise from the positive x-axis, and cos(θ) is the x-coordinate of the point you land on. In a right triangle the same ratio shows up as adjacent over hypotenuse, but the circle definition is the one that keeps working past 90°, where no triangle survives to draw. That single geometric picture explains both of cosine's defining traits: it can never exceed 1 in size, because no point on a unit circle sits farther than one unit from the center, and it repeats every full turn, because sweeping 360° further lands on exactly the same point.
Cosine and sine are the same curve, offset by a quarter turn: cos(θ) = sin(θ + 90°). Read on the circle, sine tracks the y-coordinate while cosine tracks the x-coordinate of that same moving point, so as one rises the other falls in a fixed, predictable relationship. One asymmetry is easy to miss: cosine is an even function, cos(−θ) = cos(θ), so reversing the direction of the sweep changes nothing, while sine is odd, sin(−θ) = −sin(θ), and flips sign. That single difference in symmetry is why cosine graphs start at a peak and sine graphs start at zero.
Near θ = 0 the curve flattens out rather than crossing through it, which is why the small-angle approximation cos(θ) ≈ 1 − θ²⁄2 (θ in radians) works so well for tiny angles — engineers lean on it to simplify pendulum and optics equations. At the far end of the domain there is no far end at all: θ can grow without limit, positive or negative, and cos(θ) simply keeps orbiting through the same range of values forever, never settling and never escaping [−1, 1].
- Type your angle into the Angle, θ field — the unit selector defaults to degrees.
- Switch the unit to radians or turns if that's how your angle was given; the sheet converts internally before evaluating.
- Read cos(θ) for the result — it always lands between −1 and 1, and the sign tells you which side of the vertical axis the angle points toward.
- Step the angle through 0°, 90°, 180°, and 270° to watch cos(θ) move from 1 to 0 to −1 and back to 0, tracing one full period.
Worked example — the 60° reference angle
Set the Angle, θ field to 60 with the unit selector on degrees. Internally the sheet converts that to 1.0471975511965976 radians — exactly π ⁄ 3 — before evaluating, and cos(θ) returns 0.5 on the nose, with no rounding involved, because 60° is one of the handful of angles where the ratio has a clean closed form.
The result checks itself two ways. In a 30-60-90 triangle with hypotenuse 2, the side adjacent to the 60° angle measures 1, so adjacent ⁄ hypotenuse = 1 ⁄ 2 = 0.5, matching the sheet exactly. The Pythagorean identity agrees too: sin(60°) ≈ 0.8660254, and 0.8660254² + 0.5² = 0.75 + 0.25 = 1.0 exactly, confirming the point sits precisely on the unit circle.
Questions
What does cos(θ) actually measure?
It's the x-coordinate of the point where an angle θ, swept counterclockwise from the positive x-axis, meets a circle of radius 1. Equivalently, in a right triangle it's the adjacent side divided by the hypotenuse. Both descriptions agree wherever the triangle definition applies, but the circle version keeps working past 90°, where there's no triangle left to draw.
Why is cos(θ) always between −1 and 1?
Because it's a coordinate on a circle of radius 1 — the point can never sit farther than one unit from the center, so its x-coordinate can never exceed that radius. The two extremes, cos(θ) = 1 and cos(θ) = −1, occur at θ = 0° and θ = 180°, exactly opposite ends of the same horizontal diameter.
How is cosine different from sine?
They're the same wave shifted by a quarter turn: cos(θ) = sin(θ + 90°). Sine reads the y-coordinate on the unit circle while cosine reads the x-coordinate. One structural difference matters for symmetry: cosine is even, cos(−θ) = cos(θ), while sine is odd, sin(−θ) = −sin(θ) — reversing the angle flips sine's sign but leaves cosine untouched.
Is cos the same thing as cosh?
No, despite the similar name. cos(θ) is the ordinary, periodic cosine defined from a circle — it repeats every 360° and never leaves the range [−1, 1]. cosh(x), the hyperbolic cosine, comes from a hyperbola instead, via (eˣ + e⁻ˣ)⁄2; it never repeats and grows without bound as x increases. They share a family resemblance in their series expansions and little else.
What's the most common mistake when computing cos(θ) by hand?
Mixing up degree and radian mode. cos(60) evaluated in degree mode gives 0.5, but the same keystrokes in radian mode compute the cosine of 60 radians — a value near −0.95, since 60 radians is more than nine full turns around the circle. This sheet sidesteps the trap by asking which unit you mean and converting before it evaluates anything.
What is cos(θ) at the standard reference angles?
cos(0°) = 1, cos(30°) = √3⁄2 ≈ 0.8660, cos(45°) = √2⁄2 ≈ 0.7071, cos(60°) = 1⁄2 exactly, and cos(90°) = 0. These five values, paired with their sine counterparts read in reverse order, cover most geometry and physics problems without reaching for a calculator.