Cosine Calculator

cos θ = adjacent ÷ hypotenuse cos θ = the x-coordinate of the point θ lands on

θ

The angle, in degrees or radians. It can be negative and it can be larger than a full turn — only where it lands on the circle matters

adjacent

The side next to the angle θ, in a right triangle built around it: the one that is not opposite θ and is not the hypotenuse

hypotenuse

The longest side, opposite the right angle. It is always at least as long as the adjacent side, which is why the ratio never goes above one

Reach for this page when you want one number rather than three: the cosine of a phase angle you are about to resolve into components, the fraction of a force that acts along a surface, the x-coordinate of a point you are rotating. The cosine values table below covers twelve angles, four to a turn, and it is the fastest way to see the whole shape of the function — it starts at one, falls to zero at a right angle, reaches minus one at half a turn and returns. If you need the sine and the tangent of the same angle as well, the trigonometry page prints all three together and the numbers agree to the last decimal; if what you want to picture is the point itself, the unit circle page is built around it. And if you have a cosine and want the angle back — a cosine of 0.5 belongs to more than one angle, so that is a different question with a set of answers rather than one — the inverse cosine page does that.

1. An angle of 60 degrees

   1. Rotate 60 degrees anticlockwise from the positive x-axis
   2. The point you land on is (1/2, √3/2)
   3. The x-coordinate is the cosine, so the cosine is 1/2 — the √3/2 belongs to the sine

   The angle the page loads with, and the one worth comparing against the trigonometry page: there the same angle gives the cosine 0.5 and the sine 0.866, and 0.5 is exactly what you get here. The pair is the clearest reminder that the cosine of sixty and the sine of thirty are the same number, because sixty and thirty are the two angles of one right triangle.

2. An angle of 0 degrees, the largest a cosine gets

   1. Zero degrees is no rotation at all: the point is still on the positive x-axis
   2. Its coordinates are (1, 0)
   3. The x-coordinate is 1, so the cosine is 1

   One is the ceiling. The cosine cannot exceed it at any angle, because it is the x-coordinate of a point one unit from the origin and no such point is further right than one. It reaches that value at zero degrees and again at every whole turn after it — three hundred and sixty, seven hundred and twenty — and nowhere in between.

3. An angle in the second quadrant, 120 degrees

   1. One hundred and twenty degrees is a third of a turn, so the point is in the second quadrant
   2. Its x-coordinate is now to the left of the origin, so it is negative
   3. The size of the reading is the same as at sixty degrees, and the sign has flipped

   Past a right angle the point crosses to the left of the y-axis and the cosine goes negative, where it stays until two hundred and seventy degrees. This is the single most useful thing the sign of a cosine tells you: a negative cosine means the direction you are describing points backwards, whatever the sine is doing.

4. A negative angle, minus 60 degrees

   1. A negative angle turns clockwise instead of anticlockwise
   2. The point lands below the x-axis, at (1/2, −√3/2)
   3. Its x-coordinate is the same 1/2, so the cosine is unchanged

   The cosine is an even function: it gives the same answer for an angle and for its negative. The sine and the tangent do not — they both change sign here. In a table with a single column this is easy to see, which is why the twelve angles below include two that are negatives of the ones opposite them.

Two things to keep in mind. The readings are given to four decimal places, and for most of the angles in the table the exact value is a root or a fraction that no decimal finishes: the cosine of thirty degrees is root three over two, and 0.866 is where that lands after rounding, not the number itself. The angles at which the value is exact — zero, sixty, ninety, one hundred and twenty, one hundred and eighty — are the ones where the reading looks clean, and that is a property of those angles rather than of the calculator. Second, this page does not go the other way. A cosine of 0.5 belongs to sixty degrees, to three hundred, and to infinitely many angles beyond those, because the function repeats every full turn and mirrors about zero; recovering a single angle from a cosine needs an extra decision about which range you want, and that is the inverse cosine page's job.

Is there any angle whose cosine does not exist?

No. The cosine is defined at every angle, including the ones where the tangent has no value: at ninety degrees the cosine is zero and at two hundred and seventy degrees it is zero again, both perfectly ordinary numbers. That is the difference between the two functions in one sentence — the tangent is a ratio that can be divided by zero, and the cosine never is. This page therefore never refuses a reading.

Why is the cosine of 60 degrees the same as the sine of 30 degrees?

Because sixty and thirty are the two acute angles of one right triangle. In that triangle the side next to the sixty-degree angle is the side across from the thirty-degree angle, so the ratio that gives the cosine of one is the ratio that gives the sine of the other. The table below shows the same thing from the circle side: the cosine column and the sine column are the same numbers, shifted a quarter turn.

What does a negative cosine mean?

That the direction you are describing points to the left of the y-axis — more than a quarter turn from where you started, and less than three quarters. It is not an error and it is not a smaller value: minus a half is larger than minus one, and the cosine reaches its smallest value at half a turn. Between ninety and two hundred and seventy degrees every cosine is negative, and outside that stretch every cosine is positive.

Can I enter an angle in radians?

Yes — there is a unit selector beside the box, and the angle is read the same way whichever you choose. Sixty degrees and 1.0472 radians give the same cosine, a half. The only thing to watch is which unit a source you are comparing against assumes, since by default most of them mean degrees.

Why does the cosine of 90 degrees show as zero when the sine of the same angle is exactly one?

Because at a right angle the point is on the y-axis: its x-coordinate is zero and its y-coordinate is one. The cosine is the x-coordinate, so it is zero, and the sine is the y-coordinate, so it is one. Both are exact, and the pair is the clearest illustration of what the two functions are — the cosine measures how far right the point is, and at that angle it is not to the right at all.

Why does this page give one number where the trigonometry page gives three?

Because that is what the two are for. The trigonometry page exists so the three ratios can be compared with each other, and its table has a column for each. This page exists for the reader who wants the cosine and nothing else, and its single column is what makes the shape of the function visible: one at zero, zero at a right angle, minus one at half a turn, zero again at three quarters, back to one. The two pages agree to the last decimal at every angle, so the choice is about presentation, not about the arithmetic.

• Cosine — the function the unit circle reads off as the x-coordinate, its even symmetry, and the values it takes at the special angles — Wolfram MathWorld (United States)
• Trigonometric Functions — the definitions of sine, cosine and tangent, their ranges, and why the cosine is defined everywhere while the tangent is not — Wolfram MathWorld (United States)
• Unit Circle — the circle of radius one, and why the coordinates of a point on it are the cosine and the sine — Wolfram MathWorld (United States)
• Sine, Cosine and Tangent — a first-principles walk through the three ratios in a right triangle, with the special angles worked out rather than asserted — Math is Fun (United Kingdom)