Enter an angle in degrees or radians and get its cosine instantly, along with the reciprocal secant (sec) — undefined cases clearly labeled.
Enter an angle to see its cosine and secant
This cos calculator finds the cosine of any angle you enter, in degrees or radians, and shows the reciprocal ratio, secant (sec), right alongside it. Cosine is the trigonometric ratio behind the horizontal component of rotational and periodic motion — it produces a valid value between -1 and 1 for every possible angle, making it a natural companion to sine for describing anything that moves in a circle or oscillates smoothly. Its reciprocal, secant, is undefined exactly where cosine equals zero (90°, 270°, and their multiples), and this calculator catches that condition and reports "Undefined" instead of an error.
Cosine is defined on the unit circle as the x-coordinate of the point reached after rotating by angle θ from the positive x-axis — equivalently, in a right triangle, cosine is the ratio of the side adjacent to an acute angle to the hypotenuse. This calculator converts your angle to radians internally, evaluates cos(θ) directly, then computes sec(θ) = 1/cos(θ) with a check for the zero-denominator case.
This tool is built for trigonometry and precalculus students checking cosine homework, physics students resolving the horizontal component of a force or projectile's motion, electrical engineers analyzing phase-shifted signals and power factor angles, navigators and surveyors computing horizontal distances from a bearing, and anyone who needs a quick, reliable cosine value without digging out a scientific calculator.
Cosine describes the horizontal half of circular and periodic motion, complementing sine's vertical half. Because cosine's output is always bounded between -1 and 1, and it starts at its maximum value (1) rather than zero, it's the natural choice whenever a system's motion or phase is measured from its starting peak rather than its starting center point.
Physicists split projectile motion into a cosine-based horizontal component and a sine-based vertical one. Electrical engineers use cosine in power factor calculations and AC phase-angle analysis. Navigators and surveyors compute horizontal (easting) distances from a bearing angle. Architects use cosine to calculate horizontal spans from roof pitch angles. Computer graphics and animation use cosine alongside sine for smooth circular and rotational motion.
How cosine is defined, and its reciprocal relationship with secant
Cosine never exceeds 1 or drops below -1 — its output is always bounded, mirroring sine's behavior but 90° out of phase.
Unlike tangent or cotangent, cosine has no asymptotes — every real angle produces a valid cosine value.
Secant flips cosine upside down (sec = 1/cos), so it is undefined exactly where cosine equals zero.
From entering an angle to reading the cosine and secant
Type the angle value into the input field — any real number, positive or negative, is accepted.
Select which unit your angle is measured in using the toggle above the input.
The calculator converts your angle to radians internally and evaluates cosine.
See the cosine result, always between -1 and 1, displayed in the results panel.
Review the reciprocal secant value, which is labeled "Undefined" whenever cosine equals zero.
Finding the cosine and secant of a 60° angle
A support beam makes a 60° angle with the horizontal. What is the cosine of that angle, and what is its reciprocal secant?
Undefined case: at 90°, cos 90° ≈ 0, so sec 90° is Undefined — dividing 1 by 0 has no defined result. The calculator shows this explicitly instead of displaying "Infinity."
Domain, range, and period of cosine and secant
| Function | Domain | Range | Period |
|---|---|---|---|
| cos θ | All real angles | [-1, 1] | 360° |
| sec θ | All angles except 90°, 270°, ... | (-∞,-1] ∪ [1,∞) | 360° |
Reading the result: the cosine value tells you the x-coordinate of your angle on the unit circle, scaled to a maximum swing of 1 in either direction. The secant flips that value upside down, so small cosine values near zero produce very large secant values, and vice versa.
Typical ranges: cosine stays between -1 and 1 for every angle; secant can never fall between -1 and 1 — it jumps straight from 1 (or -1) out toward infinity as cosine approaches zero.
Manual verification: for common reference angles (0°, 30°, 45°, 60°, 90°), cosine takes the clean values 1, √3/2 ≈ 0.8660, √2/2 ≈ 0.7071, 0.5, and 0 — a quick way to check the calculator's output by hand.
Where finding a cosine value is genuinely useful
Check cosine values for right-triangle and unit-circle problems.
Find the horizontal velocity or distance component of a launched object.
Compute power factor and phase-shift relationships in AC circuits.
Find the horizontal distance component from a bearing angle.
Find the horizontal span of an incline given its angle and length.
Drive circular motion or rotation alongside sine for the x-axis component.
Compute the horizontal projection of an orbital or elevation angle.
Model phase-shifted waveforms that start at a peak rather than zero.
Solve joint angle equations that use cosine alongside sine.
Break any vector into its horizontal (cosine) component.
Determine horizontal load components across angled support members.
Get an instant cosine value for any angle without a textbook or table.
What this cos calculator does well, and where it doesn't apply
How cosine compares to sine and tangent at 0°, 30°, 45°, 60°, and 90°
| Angle | cos | sin | tan |
|---|---|---|---|
| 0° | 1 | 0 | 0 |
| 30° | 0.866025 | 0.5 | 0.577350 |
| 45° | 0.707107 | 0.707107 | 1 |
| 60° | 0.5 | 0.866025 | 1.732051 |
| 90° | 0 | 1 | Undefined |
Summary: This cos calculator gives you an instant, free way to find the cosine of any angle plus its reciprocal secant, with undefined cases clearly flagged. Pair it with the Sin Calculator and Tan Calculator for a fuller trigonometry toolkit.
Common questions about the cosine function
Trusted educational references to go deeper on the cosine function
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