Cosine Calculator
Find the cosine of any angle in degrees or radians instantly, with the secant too — for triangles and vectors.
cos(60°) = 0.5
cos(60°)
0.5
Secant (1/cos)
2
Only defined when cos θ ≠ 0
Cosine is the ratio of the adjacent side to the hypotenuse in a right triangle. It reaches 1 at 0° and −1 at 180°.
Last updated: August 2026
How this calculator is verified
Checked by True Calculator automated test suite on
- Formula verified against a published worked example in the automated test suite
- Edge cases (zero, negative, boundary and unit-mismatch inputs) covered by unit tests
The full verification method is on our how we verify page. Found an error? Tell us and we will re-check it.
When to Use This Calculator
Cosine handles the horizontal side of angled problems. Engineers in India use it to resolve forces on guy wires and crane cables, where the horizontal pull equals tension times cos(angle). Ramp designers compute the floor space a wheelchair or vehicle ramp will occupy: a 5 m ramp at 30° needs about 4.33 m of floor run, since cos(30°) ≈ 0.8660. Roofers use it to convert sloped roof length into horizontal span for drainage planning, and cloth cutters use it when estimating the reduced width of fabric cut on the bias. Sailors and anglers estimate the drift of a boat pulled at an angle, and AC installers account for the cosine of the sun's angle when sizing shading. Physics students apply cosine constantly for force components and work done at an angle, making this a fixture of board-exam preparation.
How to Use This Calculator
- Step 1: Choose whether your angle is in degrees or radians.
- Step 2: Enter the angle, such as 60 for 60 degrees.
- Step 3: Read the cosine value, which always lies between −1 and 1.
- Step 4: The secant (1 divided by cosine) is shown alongside when it is defined.
Worked Example
A rickshaw puller in Varanasi pulls a load with a rope at 60 degrees to the ground. The cosine of 60° is exactly 0.5, so half of the pulling force moves the load horizontally — the effective pull equals 0.5 times the rope tension. For a ramp in a Chennai office building inclined at 30 degrees, cos(30°) ≈ 0.8660 means the horizontal run is about 86.6% of the ramp length: a 4 m ramp covers about 3.46 m of floor. The calculator confirms cos(0°) = 1, so a horizontal rope transfers its full force.
Tips and Common Mistakes
- •Tip 1: Cosine gives the horizontal component, sine the vertical — pick the one that matches the direction you need.
- •Tip 2: Cosine falls from 1 at 0° to −1 at 180°, so an angle above 90° gives a negative value, meaning the component reverses direction.
- •Tip 3: cos(θ) = sin(90° − θ), so cos(60°) and sin(30°) are both 0.5 — a quick cross-check.
- ✗Mistake 1: Using cosine for the vertical component — that is sine's job.
- ✗Mistake 2: Forgetting that negative cosines are meaningful and occur whenever the angle exceeds 90°.
Frequently Asked Questions
What is the cosine of an angle?
In a right triangle, cosine is the ratio of the side adjacent to the angle to the hypotenuse. cos(60°) = 0.5, so the adjacent side is half the hypotenuse.
What is the range of the cosine function?
Cosine returns values between −1 and 1 for every angle. It starts at 1 for 0°, falls to 0 at 90°, reaches −1 at 180° and returns to 1 at 360°.
How is cosine different from sine?
For the same angle, sine uses the opposite side while cosine uses the adjacent side. They are related by cos(θ) = sin(90° − θ), so cos(60°) equals sin(30°), both 0.5.
When would I use cosine in real life?
For horizontal components: the horizontal pull of a rope at an angle, the distance a ramp covers at the base, or the eastward component of a flight path. cos(60°) = 0.5 means half the length is horizontal.
What is secant?
Secant is the reciprocal of cosine: 1/cos(θ). Since cos(60°) = 0.5, secant is 2. It is undefined when cosine is 0, such as at 90° and 270°.
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