Tangent Calculator

tan θ = opposite ÷ adjacent tan θ = sin θ ÷ cos θ tan θ = slope

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. The tangent is the first over the second — the hypotenuse plays no part at all

sin θ ÷ cos θ

The same ratio in terms of the other two. This is the form that explains the poles: wherever the cosine is zero the division has no answer, and the cosine is zero at ninety and two hundred and seventy degrees

slope

What the ratio means on the unit circle: the gradient of the line from the origin to the point the angle lands on. A tangent of one is a forty-five degree line, and the larger the reading the steeper the line

Use this page when the quantity you want is a gradient rather than a height — the pitch of a roof quoted as rise over run, the grade of a road as a percentage, the angle a ladder or a ramp has to be set at, the direction of a line through two points. It is also the page to reach for when you want to know how steep something is in one number rather than two, since the sine and the cosine each tell you only half of the direction. Two things to keep in mind when reading it. If the angle is past ninety degrees the reading goes negative and keeps climbing in size on the other side, and if you are working from a picture rather than from coordinates, that sign is telling you the line leans the other way rather than that the steepness changed. And if the angle approaches a right angle the reading grows without bound, which is the page telling you the line has become vertical — at exactly ninety degrees there is no reading to give, because a vertical line has no run to divide by. If you want the sine and the cosine beside it, the trigonometry page prints all three; if you want the reciprocal of this number, that is a different function with its poles in the other two places.

1. An angle of 45 degrees, the one whole-number answer

   1. At forty-five degrees the two legs of the right triangle are equal
   2. The tangent is the opposite side over the adjacent side
   3. Equal quantities divided give exactly one, so the reading is 1 and not 0.9999

   The angle the page loads with, and the only angle at which this function produces an integer. Forty-five degrees is also the angle whose line has a slope of one, which is the same statement in the language of gradients: one unit up for one unit across. Every other reading on this page is either a root or the reciprocal of one, which is why the table below is written to four decimal places.

2. An angle of 30 degrees

   1. The opposite side at thirty degrees is half the hypotenuse
   2. The adjacent side is the other leg, which is half of the square root of three
   3. Half divided by half of the square root of three is one over the square root of three, about 0.5774

   The tangent of thirty degrees is the cotangent of sixty degrees, and that is a rule rather than a coincidence: the two acute angles of a right triangle add to ninety, and the side across from one is the side next to the other. So the tangent of an angle is always the cotangent of its complement, and the reciprocal statement is the other half of the same fact.

3. An angle in the second quadrant, 135 degrees

   1. One hundred and thirty-five degrees is forty-five degrees past a right angle
   2. The point is in the second quadrant, so the adjacent side is now negative and the opposite side is still positive
   3. A positive divided by a negative is negative, so the tangent is −1

   Exactly the negative of the forty-five degree reading, and that is the pattern for the whole second quadrant: the size of the tangent is unchanged from the first quadrant angle folded onto it, and only the sign moves. This is also the case the sine page reports the other way round — at one hundred and thirty-five degrees the sine is still positive at 0.7071, because the point is above the axis; the tangent is negative, because the run has crossed to the left.

4. An angle of 210 degrees, a half turn past thirty

   1. Two hundred and ten degrees is thirty degrees plus a straight angle
   2. Adding one hundred and eighty degrees flips both the opposite and the adjacent side, so both are negative
   3. A negative divided by a negative is positive, and the sizes are unchanged, so the reading is the same 0.5774

   The clearest demonstration of the half-turn period, and the place this function parts company with the sine and the cosine: those two need a whole turn to come back to the same reading, and this one is already there after half of one. The reason is that a ratio does not care when the two quantities in it both change sign. The table below shows the same pairing on rows eight apart.

Three limits worth stating plainly. The readings are given to four decimal places and most of them are irrational: the tangent of thirty degrees is one over the square root of three, so 0.5774 is a truncation rather than the answer, and forty-five degrees is the only row on this page with an exact, terminating value. Second, the tangent has no value at ninety degrees, at two hundred and seventy degrees, and at every angle that reduces to either — negative angles and angles past a whole turn included, so minus ninety and four hundred and fifty are refused for the same reason as ninety. Where a neighbouring reading would be enormous, this page gives an error message rather than a very large number, because the sequence does not converge to anything: the values grow without bound as the angle is approached from one side and fall from the bottom of the range as it is approached from the other. Third, the page does not invert. Given a tangent of one, the angle could be forty-five degrees, two hundred and twenty-five degrees, or any of the infinitely many angles a half turn apart, and choosing between them needs a decision this page does not make.

Why does the tangent have no value at 90 degrees?

Because the tangent is the opposite side divided by the adjacent side, and at a right angle the adjacent side has shrunk to nothing. Dividing by zero has no answer, so there is nothing to print. It is not that the value is enormous — the readings do grow without bound as the angle gets close to ninety, but at ninety itself there is no number, and the two sides of the approach do not even head for the same place. Two hundred and seventy degrees is the same failure half a turn later, and every angle that reduces to either of them is refused as well.

Is the tangent just the sine divided by the cosine?

Yes, and that is the shortest description of it. It is also the opposite side over the adjacent side and the slope of the line the angle draws, and all three say the same thing. The form that explains the most is the division: wherever the cosine is zero the fraction has no answer, and the cosine is zero at ninety and two hundred and seventy degrees — which is exactly where the table below has its two dashes.

Why is the tangent of 135 degrees negative when the sine is positive?

Because the tangent looks at the run as well as the rise. At one hundred and thirty-five degrees the point is still above the horizontal axis, so the sine stays positive at 0.7071, but it has crossed to the left of the vertical axis, so the adjacent side has become negative and the division turns negative with it. The sign of the tangent is really a statement about whether the two legs agree in direction, not about how high the point is — which is why it turns negative a quarter turn earlier than the sine does.

What does a very large tangent mean?

That the line is nearly vertical. The reading is a slope, so a tangent of ten means ten units up for every one across, and it grows without limit as the angle approaches ninety degrees from below. On the other side of ninety the same line reappears at the bottom of the range and climbs back towards zero, so the values on the two sides of the pole are not close to each other at all — they are as far apart as a number can be.

How often does the tangent repeat?

Every half turn, which is twice as often as the sine and the cosine. Adding one hundred and eighty degrees to an angle flips the sign of both the opposite side and the adjacent side, and a ratio of two flipped quantities is unchanged, so thirty degrees and two hundred and ten degrees share a tangent of 0.5774. The sine and the cosine, by contrast, need a whole turn to come back to the same value.

Can the angle be entered in radians?

Yes — the box has a unit selector beside it and the reading is the same angle either way. Forty-five degrees and 0.7854 radians give the same tangent, one. As always, the thing to check is which unit a source you are comparing against assumes, because most printed tables are written in degrees.

• Tangent — the opposite-over-adjacent ratio, its period of half a turn, and its poles at ninety and two hundred and seventy degrees — Wolfram MathWorld (United States)
• Trigonometric Functions — the definitions, ranges and domains of all six ratios, including the angles at which each of them is undefined — Wolfram MathWorld (United States)
• Cotangent — the reciprocal of this function, whose own poles sit at zero and one hundred and eighty degrees instead — 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)