Why does the tangent have no value at 90 degrees?

Because the tangent is the sine divided by the cosine, and at ninety degrees the cosine is zero. The point has moved to the top of the circle, its x-coordinate is zero, and there is nothing to divide by. The honest way to say it is that the tangent there has no value — not that it is very large. The sine and the cosine at that angle are both perfectly well defined, one and zero, and the table below still lists them.

Do I type the angle in degrees or in radians?

Either — 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 three numbers to four places. The difference only shows up when you compare against a source that assumes one unit, which is most textbooks: a page that says the sine of thirty is a half is talking about degrees.

What is the trigonometry table below for?

Twelve angles, four to a turn, with all three functions on each row. Between them the twelve cover every sign combination the three functions can take, both of the tangent's two poles, and all four quadrants. Read down one column at a time and the pattern of each function is visible: the sine climbs to one and comes back down, the cosine is the same shape shifted, and the tangent repeats every half turn instead of every full one.

Can the angle be negative, or more than 360 degrees?

Yes to both, and neither is a special case. A negative angle turns clockwise instead of anticlockwise, which lands on the same point as its positive counterpart added to a full turn: minus thirty degrees and three hundred and thirty degrees are the same angle and give the same three readings. Anything past a full turn keeps going around — three hundred and ninety is thirty again. Only the final position matters, so the calculator reduces the angle first and then reads the trigonometric ratios off it.

Is there a difference between this page and the unit circle page?

They share a field and they agree to the last decimal, on purpose — they are meant to be checked against each other. The difference is what they show. The unit circle page is built around the picture: a point on a circle of radius one, its coordinates, and the special angles with their exact values. This page is built around the functions and the angles that are awkward to draw: negative angles, angles past a full turn, and the two angles where the tangent has no value at all.

Why four decimal places and not more?

Because four is enough to tell a textbook answer from a wrong one, and past that the digits stop being about the angle. The exact values at most of the angles in the table are square roots and fractions, and their decimal expansions neither end nor repeat — the cosine of thirty degrees is root three over two, and 0.8660254037844386 is a truncation of it, not a different number. Printing more digits would suggest precision that the angle does not have.

Trigonometry Calculator

Result

0.5000

Sine

Cosine: 0.8660
Tangent: 0.5774

A trigonometry calculator takes one angle and gives back its sine, cosine and tangent. Type the angle, say whether it is in degrees or radians, and the page returns all three at once — the three numbers that describe where that angle lands. The sine and the cosine are two coordinates of one point. Draw a circle of radius one with its centre at the origin, start on the positive x-axis and rotate anticlockwise through your angle: the point you come to rest on has an x-coordinate and a y-coordinate, and those coordinates are the cosine and the sine. The tangent is the ratio between them, sine divided by cosine, which is also the slope of the line running from the origin out to that point. Because the tangent is a ratio rather than a coordinate it does something the other two never do: it runs out. Where the cosine is zero — at a right angle, and at three-quarters of a turn — there is nothing to divide by, so the tangent there is not a very large number, it is no number at all. This page refuses those angles and names them, rather than printing a huge value and leaving you to guess that it means the same thing. The table below sets out all three at twelve angles, four to a turn, so the pattern is visible instead of taken on trust. Angles past a full turn are ordinary input here: three hundred and ninety degrees is the same angle as thirty, and the page reads it that way. Everything below is shown to four decimal places — enough to check against a textbook, and honest about the fact that the exact answers here are roots and fractions that no decimal can finish.

Sine, cosine and tangent at twelve angles around the circle

θ (degrees)	sin θ	cos θ	tan θ
0	0	1	0
30	0.5	0.866	0.5774
45	0.7071	0.7071	1
60	0.866	0.5	1.7321
90	1	0	—
120	0.866	-0.5	-1.7321
135	0.7071	-0.7071	-1
150	0.5	-0.866	-0.5774
180	0	-1	0
210	-0.5	-0.866	0.5774
270	-1	0	—
330	-0.5	0.866	-0.5774

Twelve angles, four to a turn, and the two dashes in the tangent column are the point of the table: at ninety and at two hundred and seventy degrees the tangent has no value, and the page says so with a dash rather than with a large number. Read the sine column from the top and it rises from zero to one over the first quarter turn, falls back to zero by the half turn, goes negative, and returns — a full wave every turn. The cosine column is the same wave a quarter turn earlier. The tangent is different in kind: it repeats every half turn rather than every full one, which is why the four rows from zero to one eighty already contain its whole story. Notice also where the two positive columns peak: the sine reaches one at ninety exactly where the cosine reaches zero, and the tangent is the ratio of those two, which is what makes that angle the pole.

Formula

sin θ = opposite ÷ hypotenuse cos θ = adjacent ÷ hypotenuse tan θ = sin θ ÷ cos 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 / adjacent: The two legs of a right triangle with the angle θ in it: the side across from θ and the side next to it. Over a hypotenuse of one they become the two coordinates that the sine and the cosine read off
hypotenuse: The longest side of a right triangle, the one opposite the right angle. Dividing by it is what keeps the sine and the cosine between minus one and one

This page answers one question three ways: given an angle, what are its sine, cosine and tangent? Reach for it when you have the angle already — a roof pitch you know in degrees and want as a slope, a phase you know in radians, a direction you are about to break into components — and you want the three trigonometric ratios together rather than one at a time. The table of trig function values at twelve angles is there for the same reason: most of the angles anyone actually meets are one of those twelve, and having all three functions side by side is how you notice that the sine of thirty and the cosine of sixty are the same number. Reach for the unit-circle page instead when what you want to picture is the point itself, and for the right-triangle page when what you know is two sides rather than an angle. And if you have a ratio and want the angle back, that is the inverse direction — the sine, cosine and tangent of an angle do not tell you the angle, and the inverse pages do that job.

Worked examples

An angle of 30 degrees
1. The angle is in the first quadrant, so all three readings are positive
2. The point at 30 degrees on the circle of radius one is (√3/2, 1/2)
3. So the cosine is √3/2 and the sine is 1/2, and the tangent is their ratio, √3/3
The angle the page loads with, and the first row of the table below. Exact values are a half, root three over two and root three over three; the decimals shown are those three rounded to four places, and 0.5774 is the one worth noticing — a third of a root, not a repeating decimal.
An angle of 45 degrees
1. At 45 degrees the point is on the line y = x, so the two coordinates are equal
2. Equal coordinates mean the sine and the cosine are the same number, √2/2
3. A ratio of two equal numbers is exactly 1, which is why the tangent is exact here
The one angle below ninety degrees whose tangent is exact — 1, not 0.9999. The sine and the cosine are both 0.7071, and that is the largest either of them reaches in the first quadrant after 45: past this angle the sine keeps climbing and the cosine starts falling. This is also the only angle where the tangent equals the sine divided by the cosine and the answer comes out clean.
An angle in the second quadrant, 135 degrees
1. 135 degrees is 45 degrees past a right angle, so the point is in the second quadrant
2. Its y-coordinate is still positive, so the sine is still 0.7071
3. Its x-coordinate is now negative, so the cosine is −0.7071
4. The tangent is the ratio of a positive to a negative, so it is negative
The signs are what this example is for. The sine of 135 degrees is exactly the sine of 45, the cosine has flipped sign, and the tangent has flipped with it. Reading the table below downwards, the sine column is positive for the first two quadrants and negative for the last two; the cosine column changes at ninety; and the tangent column changes at ninety as well, which is the same place it runs out.

Limitations

Three things this page does not do. It reads an angle and gives ratios; it does not go the other way. If you have 0.5 and want the angle whose sine that is, nothing here will tell you — the inverse pages do that, and they are a genuinely different question with a genuinely different answer set, because a sine of 0.5 belongs to more than one angle. Second, the tangent of a right angle or three-quarters of a turn is refused rather than approximated, and the same refusal covers every angle that reduces to one of those two, including negative ones and ones past a full turn. That is the honest treatment, but it does mean an angle of 450 degrees returns an error where 450 is a perfectly reasonable angle for the sine and the cosine. Third, the readings are given to four decimal places. For the twelve angles in the table that is not enough to see the exact answers — the cosine of thirty is 0.866 here and √3/2 exactly — and no decimal expansion is enough, because the exact values at those angles are irrational. The table below prints the decimals; if you need the roots themselves they are in the standard exact-value chart.

Frequently asked questions

Why does the tangent have no value at 90 degrees?: Because the tangent is the sine divided by the cosine, and at ninety degrees the cosine is zero. The point has moved to the top of the circle, its x-coordinate is zero, and there is nothing to divide by. The honest way to say it is that the tangent there has no value — not that it is very large. The sine and the cosine at that angle are both perfectly well defined, one and zero, and the table below still lists them.
Do I type the angle in degrees or in radians?: Either — 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 three numbers to four places. The difference only shows up when you compare against a source that assumes one unit, which is most textbooks: a page that says the sine of thirty is a half is talking about degrees.
What is the trigonometry table below for?: Twelve angles, four to a turn, with all three functions on each row. Between them the twelve cover every sign combination the three functions can take, both of the tangent's two poles, and all four quadrants. Read down one column at a time and the pattern of each function is visible: the sine climbs to one and comes back down, the cosine is the same shape shifted, and the tangent repeats every half turn instead of every full one.
Can the angle be negative, or more than 360 degrees?: Yes to both, and neither is a special case. A negative angle turns clockwise instead of anticlockwise, which lands on the same point as its positive counterpart added to a full turn: minus thirty degrees and three hundred and thirty degrees are the same angle and give the same three readings. Anything past a full turn keeps going around — three hundred and ninety is thirty again. Only the final position matters, so the calculator reduces the angle first and then reads the trigonometric ratios off it.
Is there a difference between this page and the unit circle page?: They share a field and they agree to the last decimal, on purpose — they are meant to be checked against each other. The difference is what they show. The unit circle page is built around the picture: a point on a circle of radius one, its coordinates, and the special angles with their exact values. This page is built around the functions and the angles that are awkward to draw: negative angles, angles past a full turn, and the two angles where the tangent has no value at all.
Why four decimal places and not more?: Because four is enough to tell a textbook answer from a wrong one, and past that the digits stop being about the angle. The exact values at most of the angles in the table are square roots and fractions, and their decimal expansions neither end nor repeat — the cosine of thirty degrees is root three over two, and 0.8660254037844386 is a truncation of it, not a different number. Printing more digits would suggest precision that the angle does not have.

References

Trigonometric Functions — the definitions of sine, cosine and tangent, their domains, and why the tangent is undefined where the cosine is zero — Wolfram MathWorld (United States)
Sine — the function the unit circle reads off as the y-coordinate, its period, and the values it takes at the special angles — Wolfram MathWorld (United States)
Tangent — the ratio the other two functions produce, its period of half a turn rather than a whole one, and its poles at ninety and two hundred and seventy degrees — 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)

Related calculators

Unit Circle CalculatorSine, cosine and tangent of an angle, read off the unit circle.Right Triangle CalculatorSolve a right triangle from two sides or a side and angle.Cosine CalculatorCosine of an angle in degrees or radians.Cotangent CalculatorCotangent of an angle in degrees or radians.