Pick a calculator with the three options above the tool: Standard Trigonometry Functions, Inverse Trigonometry Functions or Angle Explorer. Learn more about switching
In the standard calculator, type the angle in the Angle box and pick Degrees or Radians in Select Units. Learn more about the angle
Pick how many decimal places the table shows in Select Precision Level; the default is 5. Learn more about precision
Click Calculate. The table fills with sin, cos, tan, csc, sec and cot of the angle; each value has a copy button. Learn more about the table
Click a blue header of the table, sin to cot, to read that function's explanation in the panel on the right; the text ends with a link to the section below that treats it in full. Learn more about the panel
Three options sit above the tool. Standard Trigonometry Functions takes an angle and returns its sine, cosine, tangent, cosecant, secant and cotangent. Inverse Trigonometry Functions takes a ratio and returns the angle whose sine, cosine or tangent it is. Angle Explorer is a visual tool for looking at an angle in standard position; a short caption above it says how to use it, and it has no explanation panel of its own.
Click an option and the tool below changes at once. Each calculator keeps its own numbers while you look at another, so you can switch back without retyping. The standard calculator opens first, with the Angle box empty and Degrees selected, so typing 30 gives the angle 30∘.
The two calculators share the same layout: the inputs and the buttons on the left, the explanation panel on the right, and the table of results underneath with a copy button beside every value.
Type the angle in the Angle box of the standard calculator. The box's placeholder names the unit in force: Enter Angle in Degrees or Enter Angle in Radians. Whole numbers, decimals and negative angles all work, and angles beyond one turn are fine: 390∘ gives the same values as 30∘.
Pick the unit in Select Units. The same angle has two names: 30∘ in degrees is π/6≈0.5236 in radians, since a full turn is 360∘ or 2π. Changing the unit does not convert the number already in the box, so after switching to radians type the radian value yourself.
A mismatch is the usual source of surprising results: 30 radians is almost five turns, and its sine is −0.98803, not 0.5. The placeholder tells you which unit the calculator will assume.
How degrees and radians are defined and converted is explained in degrees and radians.
Choosing the Precision
DemoDecimal places
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Select Precision Level sets how many decimal places the table shows. The default is 5, which prints sin45∘ as 0.70711; with 2 places it reads 0.71 and with 8 places 0.70710678. The choice affects only the display, not the calculation, and it applies to all six values at once.
Whole-number results print without decimals: sin30∘=0.50000 at five places is really 21, and csc30∘ prints as 2. A value that is undefined, such as tan90∘, prints as Undefined whatever the precision.
Use fewer places when you want to recognise an exact value and more when the result feeds another calculation; rounding too early is the usual cause of a wrong final digit. The same setting exists in the inverse calculator.
Click Calculate and the standard calculator sends the angle and the unit to the server, which returns the six values; the table under the form fills with them. For 30∘ it shows sin=0.50000, cos=0.86603, tan=0.57735, csc=2, sec=1.15470 and cot=1.73205. Until you click, every cell shows three dots. The demo above stops at the button, because the request would go out every time the demo loops.
A value that does not exist prints as Undefined: tan90∘ and sec90∘, because cos90∘=0, or csc0∘ and cot0∘, because sin0∘=0. If the server cannot be reached, the message Failed to fetch data appears instead of values.
Every cell has a copy button that puts the value on the clipboard. The headers of the table are also buttons: click one to read that function's explanation, as described in the next section.
Why some values are undefined is explained in tangent and cotangent.
The Function Headers and the Explanation Panel
DemoOne explanation per function
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The six headers of the table, sin, cos, tan, csc, sec and cot, are blue and clickable; hovering one shows the hint *Click for Explanation*. Click a header and the panel on the right shows the function's name in capitals, a short definition with the right-triangle ratio and the unit-circle reading, and a link to an external reference. Each definition ends with a link to the section on this page that treats the function in full.
The panel shows sine when the page opens and after Reset. Clicking a header does not change the numbers in the table; it only changes the explanation, so you can read all six for one angle.
The inverse calculator works the same way: its headers arcsin, arccos and arctan open the explanations of the inverse functions, with the input range and the output range of each.
The reciprocal relations the explanations mention, csc=1/sin, sec=1/cos and cot=1/tan, are set out in cosecant, secant and cotangent.
Using the Inverse Calculator
DemoArcsine, arccosine and arctangent of a ratio
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Pick Inverse Trigonometry Functions to go from a ratio back to an angle. Type the ratio in the Ratio box, choose Radians or Degrees in Angle Unit, set the precision, and click Calculate. The table shows arcsin, arccos and arctan of the ratio, computed on this page without a server.
For the ratio 0.5 in radians the table reads arcsin 0.52360, arccos 1.04720 and arctan 0.46365; in degrees, 30, 60 and 26.56505. Arcsine and arccosine accept ratios from −1 to 1 only, because a sine or cosine never leaves that range; arctangent accepts any number, since a tangent can be anything.
Type 2 and click Calculate: the message Value outside [-1,1] range: arcsin/arccos undefined, arctan calculated appears, the first two cells read Undefined, and arctan 2 is still given, 63.43495∘.
Which angle the calculator returns when several would fit is explained in the inverse functions.
Reset and Messages
DemoA message, then Reset
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Each calculator has a Reset button next to Calculate. In the standard calculator it empties the Angle box, returns the unit to Degrees and the precision to 5 places, clears the table back to dots and sets the panel to sine. In the inverse calculator it empties the Ratio box, clears the table and the message, and sets the panel to arcsine.
Messages appear in red under the form. The inverse calculator warns when the ratio lies outside −1 to 1: arcsine and arccosine are undefined there, arctangent is still computed, for example arctan 1.5=0.98279 radians. The standard calculator reports Failed to fetch data when the server does not answer; try again in a moment.
Reset clears the angle and the table in both calculators and sends nothing to the server.
The range rule behind the inverse calculator's message is explained in arcsine and arccosine.
The Six Trigonometric Functions
In a right triangle with an acute angle θ, the three basic functions are ratios of sides: sinθ is opposite over hypotenuse, cosθ adjacent over hypotenuse, and tanθ opposite over adjacent. The three others are their reciprocals: cscθ=1/sinθ, secθ=1/cosθ and cotθ=1/tanθ.
On the unit circle the definitions extend to every angle: the point at angle θ has coordinates (cosθ,sinθ), and tanθ is the slope of the radius to it. That is how the calculator evaluates sin150∘=0.5 or cos200∘=−0.93969, angles that fit in no right triangle.
A few angles have exact values worth knowing: at 30∘, sin=21, cos=23≈0.86603, tan=31≈0.57735; at 45∘, sin=cos=22≈0.70711 and tan=1. The calculator's six headers are explained one by one in sine, cosine, tangent, cosecant, secant and cotangent.
The sine of an angle is the ratio of the opposite side to the hypotenuse in a right triangle, and on the unit circle the y-coordinate of the point at that angle. It runs from −1 to 1: sin0∘=0, sin30∘=21, sin90∘=1, sin150∘=21 again, and sin270∘=−1.
Sine repeats every 360∘, or 2π radians, and is odd: sin(−θ)=−sinθ. Its graph is the familiar wave, which is why sine describes oscillations: sound, alternating current, tides. Together with cosine it satisfies sin2θ+cos2θ=1.
The common mistake is to expect sin30 to be 0.5 when the calculator is set to radians: 30 radians is almost five turns, and sin30≈−0.98803. Check the unit before reading the table.
The cosine of an angle is the ratio of the adjacent side to the hypotenuse, and on the unit circle the x-coordinate of the point at that angle. It also runs from −1 to 1: cos0∘=1, cos60∘=21, cos90∘=0, cos180∘=−1.
Cosine is the sine shifted by a quarter turn, cosθ=sin(90∘−θ), which is the cofunction relation; it is even, cos(−θ)=cosθ, and repeats every 360∘. Its zeros at 90∘ and 270∘ are where tangent and secant become undefined.
Cosine appears in the law of cosines, which generalises Pythagoras to any triangle, and in the dot product of vectors. The common mistake is to mix up which side is adjacent: it is the side that touches the angle and is not the hypotenuse.
The tangent of an angle is the ratio of the opposite side to the adjacent side, equal to sinθ/cosθ. On the unit circle it is the slope of the radius to the point at that angle, so it can take any real value: tan0∘=0, tan30∘≈0.57735, tan45∘=1, tan60∘≈1.73205.
Where the cosine is zero the tangent is undefined, at 90∘, 270∘ and every half turn beyond; the calculator prints Undefined there. Near those angles the tangent grows without bound, which gives its graph the vertical asymptotes. Tangent repeats every 180∘, half the period of sine and cosine, and is odd.
Tangent is the function of slopes and gradients: a road rising 1 metre over 10 metres has an angle whose tangent is 0.1. The common mistake is to round tan89.9∘ to infinity; it is large, about 572.96, but defined.
The cosecant is the reciprocal of the sine: cscθ=1/sinθ, the hypotenuse over the opposite side. Since ∣sinθ∣≤1, the cosecant is never between −1 and 1: csc30∘=2, csc90∘=1, csc150∘=2.
It is undefined wherever the sine is zero, at 0∘, 180∘ and every half turn, and the calculator prints Undefined there. It shares the period 360∘ and the odd symmetry of the sine, and it is large exactly where the sine is small.
The cosecant appears mainly in identities and in calculus, where 1+cot2θ=csc2θ is the companion of the Pythagorean identity. The common mistake is to confuse the cosecant with the inverse sine: cscθ is a reciprocal of a value, arcsinx is an angle; csc30∘=2 while arcsin0.5=30∘.
The secant is the reciprocal of the cosine: secθ=1/cosθ, the hypotenuse over the adjacent side. It is never between −1 and 1: sec0∘=1, sec30∘≈1.15470, sec60∘=2, sec180∘=−1.
It is undefined wherever the cosine is zero, at 90∘ and 270∘, where the calculator prints Undefined. Like the cosine it is even and repeats every 360∘. The identity 1+tan2θ=sec2θ follows from dividing sin2θ+cos2θ=1 by cos2θ.
The secant names the length of a line that cuts a circle, which is where the word comes from. The common mistake is the same as for the cosecant: secθ is not arccos, the inverse cosine; sec60∘=2 while arccos0.5=60∘.
The cotangent is the reciprocal of the tangent: cotθ=1/tanθ=cosθ/sinθ, the adjacent side over the opposite side. cot30∘≈1.73205, cot45∘=1, cot60∘≈0.57735, so the cotangent of an angle is the tangent of its complement.
It is undefined where the sine is zero, at 0∘ and 180∘, the opposite of the tangent's gaps at 90∘ and 270∘; the calculator prints Undefined at both kinds of angle for the right function. At 90∘ the cotangent is 0 although the tangent is undefined there. It repeats every 180∘ and is odd.
The common mistake is to compute 1/tanθ at an angle where the tangent is undefined and conclude the cotangent is too: cot90∘=cos90∘/sin90∘=0/1=0. Use the cosine-over-sine form at those angles.
A degree is 3601 of a full turn. A radian is the angle whose arc on a circle equals the radius, so a full turn is 2π radians and 180∘=π radians. To convert, multiply degrees by 180π or radians by π180: 30∘=6π≈0.5236 and 1 radian ≈57.2958∘.
The calculator computes with whichever unit Select Units shows, and the inverse calculator returns its angles in the unit of Angle Unit. Degrees are the everyday unit; radians are the natural one in calculus, where dxdsinx=cosx holds only for x in radians, and in formulas such as the arc length s=rθ.
The common mistake is to type a degree value while radians are selected, or the reverse: sin30 radians is about −0.98803, nothing like sin30∘=0.5. The placeholder of the Angle box names the unit in force.
The inverse functions run the ratios backwards: arcsinx is the angle whose sine is x, arccosx the angle whose cosine is x, arctanx the angle whose tangent is x. They are also written sin−1, cos−1 and tan−1, which does not mean 1/sin; that is the cosecant.
Because the sine of many angles is 0.5, the inverse must choose one: each function returns the angle in a fixed principal range. Arcsine and arctangent return angles from −90∘ to 90∘, arccosine from 0∘ to 180∘. So arcsin0.5=30∘, not 150∘, and arccos0.5=60∘. The input of arcsine and arccosine must lie between −1 and 1; arctangent takes any number.
The calculator's three headers are explained in arcsine, arccosine and arctangent. To find every angle with a given sine, take the principal value and add the symmetric partner and multiples of a full turn.
Arcsine returns the angle between −90∘ and 90∘ whose sine is the input: arcsin0.5=30∘ or 6π≈0.52360 radians, arcsin1=90∘, arcsin(−1)=−90∘, arcsin0=0.
The input must lie between −1 and 1, because no angle has a sine outside that range; the calculator reports a ratio such as 2 as undefined. Arcsine is odd and increasing, and sin(arcsinx)=x for every allowed x, while arcsin(sinθ)=θ only for θ in the principal range: arcsin(sin150∘)=30∘.
Arcsine solves for an angle from a ratio of sides: a ramp that rises 1 metre over a 4-metre slope has angle arcsin0.25≈14.48∘. The common mistake is to read sin−1x as 1/sinx.
Arccosine returns the angle between 0∘ and 180∘ whose cosine is the input: arccos0.5=60∘ or 3π≈1.04720 radians, arccos1=0, arccos0=90∘, arccos(−1)=180∘.
As with arcsine, the input must lie between −1 and 1. Arccosine is decreasing: a larger cosine means a smaller angle. It is linked to arcsine by arccosx=90∘−arcsinx, so arccos0.5=90∘−30∘=60∘, and arccos(−x)=180∘−arccosx.
Arccosine gives the angle between two vectors from their dot product and the angles of a triangle from its sides, by the law of cosines. The common mistake is to expect a negative result for a negative input: arccos(−0.5)=120∘, a positive angle in the second quadrant.
Arctangent returns the angle between −90∘ and 90∘ whose tangent is the input: arctan1=45∘ or 4π≈0.78540 radians, arctan0.5≈26.56505∘, arctan2≈63.43495∘, arctan0=0.
Any real number is a valid input, because a tangent can be any real number; that is why the calculator still returns arctan when a ratio outside −1 to 1 makes arcsine and arccosine undefined. As the input grows the angle approaches 90∘ without reaching it: arctan1000≈89.94∘. Arctangent is odd and increasing.
Arctangent turns a slope into an angle: a gradient of 0.1 is an angle of arctan0.1≈5.71∘. The common mistake is to use it for a direction without checking the quadrant: the points (1,1) and (−1,−1) both have slope 1, but only the first lies at 45∘.