Sine Calculator

sin θ = opposite ÷ hypotenuse sin θ = y ÷ r sin θ = −sin(−θ)

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

opposite / hypotenuse

The two sides a right triangle with the angle θ in it gives you: the side across from θ, and the longest side. The sine is the first over the second, which is why it is never larger than one and never smaller than minus one

y / r

The same ratio read on a circle of radius r: y is the height of the point the angle lands on. On the unit circle r is one, so the sine is simply that height

quadrant

Which quarter of the coordinate plane the angle lands in, counted anticlockwise from the positive horizontal axis. Zero means the angle sits exactly on an axis and belongs to no quadrant

Use this page when you have an angle and want the height that goes with it — the vertical component of a slope, the rise of a ramp measured against its length, the resolution of a force into its upward part. It is also the quickest way to check a sign: if you know the angle is past a right angle, the sine is still positive until one hundred and eighty degrees, and this page prints the quadrant so you can see that the point has moved left while it stayed up. The reference table puts the same twelve angles in the same order as the cosine page, so the two can be read side by side: one column there is a dash wherever the tangent gives up, and this page never has a dash at all. If you want the sine beside the cosine and the tangent, the trigonometry page prints all three together. And if what you have is a sine value and you want the angle back, that is the inverse direction, where two angles share every value between minus one and one.

1. An angle of 30 degrees, the page's own starting value

   1. Half of an equilateral triangle is the right triangle at 30 degrees
   2. The side across from the thirty degree angle is half the hypotenuse
   3. One half divided by one is 0.5, and the point is above the axis and to the right, so the quadrant is 1

   The angle the page loads with, and the one reading on this page that is exact in a way a reader can see without decimals. Thirty degrees is also where the sine is smallest among the special angles whose value is not zero — a quarter turn is one, an eighth of a turn is 0.7071, and this is 0.5. The quadrant column reads 1 because the point is above the horizontal axis and to the right of the vertical one.

2. An angle of 150 degrees, the same sine in another quadrant

   1. One hundred and fifty degrees is thirty degrees short of a straight angle
   2. The point has crossed to the left of the vertical axis but is still above the horizontal one
   3. The height is the same as at thirty degrees, so the sine is 0.5 and the quadrant has become 2

   The clearest case for why this page prints a quadrant at all: the number on its own cannot tell these two angles apart. Thirty and one hundred and fifty degrees share a sine of 0.5, and they are the only two angles in a full turn that do. That pairing is a rule rather than a coincidence — the sine of an angle equals the sine of its supplement, which is what you see when you fold the circle down the vertical axis.

3. A negative angle, minus 30 degrees

   1. Minus thirty degrees is thirty degrees measured the other way, clockwise from the positive horizontal axis
   2. The point is now below the axis rather than above it, so the height is negative
   3. The reading is −0.5 and the quadrant is 4

   This is what an odd function means: change the sign of the angle and the answer changes sign with it. The cosine does not behave this way — its reading at minus thirty degrees is the same 0.866 that it gives at plus thirty. The difference is worth holding on to, because it comes back in every formula that mixes the two, and it is why the table below is not symmetric about its first row.

4. An angle of 90 degrees, where the quadrant column reads zero

   1. At a right angle the opposite side and the hypotenuse are the same side
   2. A side divided by itself is one, so the sine is 1 — the largest value it ever takes
   3. The point is on the vertical axis rather than inside a quarter, so the quadrant is 0

   The reading of 1 is the top of the range, and no angle exceeds it. The zero in the quadrant column is the convention this page follows for all four axis angles: zero, ninety, one hundred and eighty and two hundred and seventy degrees are not in any quadrant, because a quadrant is an open quarter and the axis is its boundary. Both zeros in the table below come from that rule, and this is the row where it is easiest to see why it is the honest answer rather than a missing value.

Three limits worth stating plainly. The readings are given to four decimal places and most of them are irrational: the sine of sixty degrees is half of the square root of three, so 0.866 is a truncation rather than the answer, and thirty, ninety and their reflections are the only rows on this page with short exact values. Second, the quadrant is a label for a quarter of the plane, not a measurement, and the four angles that sit on an axis belong to no quarter at all — the column reads zero for them. That zero is not a missing value and not a fifth quadrant; it is the statement that the angle is on the boundary. Third, the page does not invert. Given a sine of 0.5, the angle could be thirty degrees, one hundred and fifty degrees, or either of those plus any whole number of turns, and choosing between them needs a decision this page does not make. Angles beyond a million degrees in either direction are refused rather than reduced, because the reduction itself stops being meaningful at that size.

Why does the sine never run out, when the tangent does?

Because the division is the other way round. The sine is the opposite side over the hypotenuse, and the hypotenuse is the longest side, so the fraction is always between minus one and one and the bottom is never zero. The tangent is the opposite side over the adjacent side, and at a right angle the adjacent side is zero — dividing by zero has no answer, so that is the angle the tangent page refuses. This page accepts every angle, including ninety degrees, where the reading is one.

What is the quadrant column telling me?

Which quarter of the coordinate plane the angle lands in, counted anticlockwise from the positive horizontal axis. The sign of the sine says whether the point is above or below the axis and the sign of the cosine says whether it is left or right, so the two signs together name the quarter. This page prints the quadrant because the sine alone cannot: 30 degrees and 150 degrees share a sine of 0.5, and only the quadrant says which of them you have.

Why do the axis angles show a quadrant of zero?

Because zero, ninety, one hundred and eighty and two hundred and seventy degrees sit on an axis rather than inside a quarter, and a quadrant is an open quarter whose boundary is the axis. Filing them under the first quadrant would be a claim that is not true — the point is not above the horizontal axis, it is on it. The zero is the honest answer, and it is the same convention the reference table uses in its quadrant column.

Why is the sine of 150 degrees the same as the sine of 30 degrees?

Because the sine of an angle equals the sine of its supplement. One hundred and fifty degrees is thirty degrees short of a straight angle, and folding the circle down its vertical axis carries one onto the other, so the two points sit at the same height. The angles that share a sine are always a pair inside one full turn, and they are mirror images about the vertical axis rather than about the horizontal one — which is exactly why the quadrant changes while the value does not.

Is the sine ever negative?

Yes, at every angle below the horizontal axis, which is the second half of the turn and every negative angle. The reading is negative between one hundred and eighty and three hundred and sixty degrees, and the smallest value it reaches is minus one, at two hundred and seventy. In this respect the sine differs from the cosine at a quarter turn rather than a half: the cosine turns negative once the point crosses the vertical axis, and the sine turns negative once it crosses the horizontal one.

Can I enter the angle in radians?

Yes — the box has a unit selector beside it and the reading is the same angle either way. Thirty degrees and 0.5236 radians give the same sine, one half, and the same quadrant. The thing to check is which unit a source you are comparing against assumes, because most printed tables are written in degrees while most programming languages are not.

• Sine — the opposite-over-hypotenuse ratio, its odd symmetry, and its period of one full turn — Wolfram MathWorld (United States)
• Trigonometric Functions — the definitions of all six ratios, their ranges, and why the sine is defined at every angle while the tangent is not — Wolfram MathWorld (United States)
• Unit Circle — the circle of radius one, the four quadrants it is divided into, and why a point on it is read as a cosine and a sine — Wolfram MathWorld (United States)
• Sine, Cosine and Tangent — a first-principles walk through the ratios in a right triangle, with the special angles worked out rather than asserted — Math is Fun (United Kingdom)