What is TrigPro? One trigonometry calculator, three tools

Trigonometry has a habit of turning up when you least expect it. One minute you are revising for a maths exam, the next you are trying to work out how long a rafter needs to be, or how far the foot of a ladder should sit from the wall. The problem underneath is nearly always the same: you know a couple of things about a triangle and you need the rest.

That is what this trigonometry calculator is for. We built it because we kept juggling three separate things: a scientific calculator for sin, cos and tan, a unit circle chart for exact values, and a triangle solver for the missing sides and angles. TrigPro, the calculator above this article, puts all three into one panel. It is a free trig calculator, there is nothing to install, and it works as comfortably on a phone as on a laptop.

Further down you will find a proper walk-through of every button, the trigonometry you need to make sense of the answers, a set of worked examples with real numbers, and a FAQ. If you only want the controls explained, jump straight to How to use this calculator. If you are a student, it is also worth keeping our grade calculator open in another tab, so you can see how a few marks on a trig question move your overall result.

Between them these three tabs cover what people usually search for: a sin cos tan calculator, a trigonometric function calculator, a unit circle calculator, an inverse trig calculator, a right triangle calculator and a general triangle calculator that handles the law of sines and the law of cosines for you.

How to use this calculator

The calculator has three tabs along the top: Calculator, Trig values and Triangle. Tap a tab and the panel slides across. Everything is built to fit on one screen, so you never need to scroll inside it, and the page jumps back to the top of the calculator whenever you press a main button. Here is what each part does.

1. The Calculator tab: a sin cos tan calculator online, free

This is a full scientific calculator that happens to be very good at trigonometry. Use it when you have an expression to evaluate rather than a triangle to solve, which makes it your quickest sin cos tan calculator online.

Step 1: pick the angle mode

Just above the display you will see DEG, RAD and GRAD. DEG means degrees (360 in a full turn), RAD means radians (2π in a full turn) and GRAD means gradians (400 in a full turn). A small badge in the top-left corner of the display always shows which mode you are in. Getting this wrong is the single most common reason for a trig answer that looks odd, so make it a habit to glance at that badge first.

Step 2: type your calculation

1. Check the badge says DEG.
2. Tap sin, type 30, tap ) and then =.
3. The display shows 0.5.

You can forgive yourself a missing closing bracket at the very end, because TrigPro closes it for you. Typing sin(30 and pressing equals still gives 0.5. Brackets in the middle of an expression do need closing, so sin(30)+cos(60) stays as it is.

What the other keys do

• 2nd turns the top rows into their partners: sin, cos and tan become sin^-1, cos^-1 and tan^-1, x² becomes x³, the square root becomes a cube root, log becomes 10^x and ln becomes e^x. After you press one of these the 2nd key switches itself off again.
• π and e insert the constants. n! is factorial, and it also copes with in-between numbers, so 0.5! gives about 0.8862.
• % turns a number into a hundredth of itself, Ans brings back your last answer, and the ° key tells the calculator that a number is in degrees even if you are in RAD mode. For example 90° in RAD mode is treated as a right angle, not as ninety radians.
• AC clears everything and the backspace key removes one entry at a time. It removes a whole function name such as sin( in one go.
• MC, MR, M+ and M− are the memory keys. M+ adds the current result to memory, M− subtracts it, MR drops the stored number into your expression and MC empties it. When memory holds something, an M badge shows the value.
• The clock icon opens your history, which keeps your last 30 calculations. Tap any entry and it reloads the expression together with the angle mode you used. There is a bright Clear history button at the bottom.
• Tap the result to copy it to your clipboard, ready to paste into homework, a spreadsheet or a message.

On a laptop you can type instead of tapping. Click the calculator once, then use the number keys, + − * / ^ ( ) ! %, the letters s, c and t for sin(, cos( and tan(, p for π, Enter or = for equals, Backspace to delete and Escape to clear.

When something cannot be calculated, the display tells you plainly. Try tan(90) in degree mode and you get Math Error, because the tangent of 90° really is undefined. The same goes for dividing by zero, or asking for the inverse sine of a number bigger than 1.

2. The Trig values tab: all six ratios for any angle

Open the Trig values tab when you want everything about one angle at once. It acts as a sine cosine tangent calculator, an exact trig values calculator and a unit circle calculator in one.

1. Make sure Find values is selected at the top left.
2. Choose DEG or RAD on the right. (The Trig values tab works in degrees and radians; gradians live on the Calculator tab.)
3. Type your angle. Negative angles and big ones such as 750 are fine. In radian mode you can type fractions of π, for example 5π/6, using the π button. The quick buttons underneath fill in common angles like 30°, 45° or π/3 with one tap.
4. Press Calculate.

You get six result cards: sin, cos, tan, csc, sec and cot. For every multiple of 15° the card shows the exact form first, such as −√3/2 for cos 150°, with the decimal underneath it. For other angles you get the decimal to 12 significant figures. Where a function does not exist, for example tan 90°, the card says Undefined rather than showing a giant number. Tap any card to copy its value.

Underneath the six cards is a row of four extras that save a lot of working out: the reference angle, the angle converted to radians (or to degrees if you typed radians), the quadrant the angle sits in, and its coterminal angles, which are the angles that finish in the same place after extra turns.

The drawing above the form is not just decoration. It sweeps round to your angle on a unit circle. The white line is the radius, the blue line is the cosine, the red line is the sine and the yellow line shows the tangent. Beside it, the sine curve is drawn out as the angle grows, so you can see where the number comes from. When you want to try another angle, press Edit angle.

3. Find angle: an inverse trig calculator

Sometimes you already know the ratio and need the angle. Switch the top selector to Find angle and this tab becomes a find angle calculator for trigonometry, handling arcsin, arccos and arctan along with their reciprocal cousins.

1. Pick the function with the small buttons: sin, cos, tan, csc, sec or cot.
2. Type the value. Decimals such as 0.5 work, and so do fractions and roots such as √3/2 or 1/3. The √ and / buttons help on a phone.
3. Press Calculate.

The answer comes in three parts. The principal value is the single angle a calculator’s inverse button would give you. Then comes the full list of angles between 0° and 360° that fit, and finally the general solution, such as θ = 30° + 360°n, where n is any whole number. Try sin θ = 0.5 and you get 30° and 150°. Try cos θ = −0.5 and you get 120° and 240°. The graph shows a dashed horizontal line crossing the curve at each solution, which makes it obvious why there are usually two answers in one turn.

If you ask for something impossible, such as sin θ = 2, it explains why. A sine can never be bigger than 1.

4. The Triangle tab: solve any triangle

This is the part that works as a solve triangle calculator, and in everyday terms it is a find missing side of a triangle calculator and a find missing angle of a triangle calculator rolled into one. Keep Any triangle selected at the top.

1. Type any three values into the six boxes, as long as at least one of them is a side. Sides are a, b and c. Angles are A, B and C. Side a sits opposite angle A, b opposite B and c opposite C.
2. Press Calculate.
3. Read the results, or press Edit values to change something, or Reset (the orange button) to clear the lot.

Those three values can be any of the classic combinations: SSS, SAS, ASA, AAS and the tricky SSA. You do not have to tell the calculator which one you are using. It works that out from what you typed, then uses the law of cosines or the law of sines as needed, so it also does the job of a law of sines calculator and a law of cosines calculator.

The result is a grid of cards, colour-coded so you can find things quickly. Values you typed are tagged given. Values the calculator worked out are shown in coloured text. You get:

• all three sides and all three angles;
• the height to each side (the perpendicular distance from the opposite corner), which makes it a handy triangle height calculator;
• the area, perimeter and semi-perimeter, so it doubles as a triangle area calculator that finds the area from three sides without you having to remember Heron’s formula;
• the inradius (the radius of the largest circle that fits inside) and the circumradius (the circle that passes through all three corners);
• the type of triangle, for example acute scalene, obtuse scalene or equilateral.

Above the cards, the triangle is drawn to scale. The outline draws itself, the corners are labelled A, B and C, and the side lengths and angles are written on it. Tap any card to copy its value.

Choosing the unit: cm, m, in or ft

At the top right of the Triangle tab there is a small switch for cm, m, in and ft. Pick the unit you are measuring in and it is added to every length in the answer, and to the area as a square unit, for example 6 cm². Be aware that it is a label, not a converter. The three sides of a triangle must be in the same unit when you type them in. If your measurements are mixed, convert them first. Our height converter is useful when you have a height in feet and inches that needs turning into metres before you start.

5. Right triangle mode

Most real-life triangle problems have a right angle in them, so this has its own switch. Tap Right triangle next to Any triangle. Angle C is locked at 90° (you will see the padlock), side c becomes the hypotenuse, and you only have to type two values, at least one of them a side. As a right triangle calculator with sides and angles, it handles every combination: two legs, a leg and the hypotenuse, or a side and one acute angle.

The drawing also changes. The right angle sits in the bottom-left corner with a small square marker, with the two legs running along the bottom and up the side, so there is no mistaking it for an ordinary triangle. If all you need is a missing length, this works as a Pythagorean theorem calculator and a hypotenuse calculator too: enter the two legs and read off side c.

It is a quick 3 4 5 right triangle calculator as well. Type 3 and 4 and you get a hypotenuse of exactly 5, with angles of 36.8699° and 53.1301°. The same mode doubles as a 30 60 90 triangle calculator (try c = 10 and angle A = 30) and a 45 45 90 triangle calculator (try a = 1 and b = 1). It is also a right angle triangle calculator with all the working values shown: the three heights, area, perimeter, inradius and circumradius, in one go.

6. The ambiguous case (SSA)

Give the Triangle tab two sides and an angle that is not between them and sometimes two different triangles fit the facts. Rather than quietly picking one, TrigPro shows both. A Triangle 1 / Triangle 2 switch appears over the drawing, and each button redraws the sketch and refreshes every result card. If you are looking for an ambiguous case triangle calculator, try side a = 7, side b = 10 and angle A = 40°. You will see two answers: angle B is 66.6742° in one and 113.3258° in the other. If no triangle can exist, or the three angles add up to 180° or more, the calculator says so in plain words instead of showing nonsense.

Trigonometry basics: SOHCAHTOA in plain English

Take any right-angled triangle and pick one of the two corners that is not the right angle. Call the angle there θ (theta). From that corner’s point of view, the three sides get names. The longest side, facing the right angle, is the hypotenuse. The side facing θ is the opposite. The side that touches θ and is not the hypotenuse is the adjacent. Swap to the other corner and opposite and adjacent swap too, which is why it pays to label the triangle before you start.

Sine, cosine and tangent are simply ratios of those sides. The standard memory aid is SOH CAH TOA. If you have ever searched for a SOHCAHTOA calculator, the Triangle tab is exactly that, working in both directions.

In that example the angle θ is the one opposite the side of length 3. The three other functions are just upside-down versions: cosecant (csc) is 1/sin, secant (sec) is 1/cos and cotangent (cot) is 1/tan. They are rarely needed in everyday work, but they appear in A-level questions, which is why the Trig values tab shows all six.

How to calculate sin, cos and tan

If you are wondering how to calculate sin cos tan for a triangle you have drawn, here is the routine that works every time.

1. Sketch the triangle and mark the right angle.
2. Label the sides as opposite, adjacent and hypotenuse from the angle you care about.
3. Decide which two sides are involved. Opposite and hypotenuse means sine, adjacent and hypotenuse means cosine, opposite and adjacent means tangent.
4. Write the ratio as a fraction and divide.
5. To go the other way, from a ratio to an angle, use the inverse button (sin^-1, cos^-1 or tan^-1).

Here is a small example. A ramp rises 1.5 m over a horizontal distance of 6 m. The opposite side is 1.5 and the adjacent side is 6, so tan θ = 1.5 / 6 = 0.25. Take the inverse tangent and θ = 14.04°. The ramp’s sloping length comes from Pythagoras: the square root of (1.5² + 6²) = 6.18 m. Put 1.5 and 6 into the Triangle tab in right triangle mode and it shows 14.0362°, 75.9638° and a hypotenuse of 6.1847 m in one go.

Nobody works out the sine of an awkward angle like 37° by hand any more. Before calculators people used printed tables, and the numbers inside them were produced by repeatedly applying identities. Today you press a key. The sine of 37° is 0.6018, and that is the kind of number the Calculator tab is for.

The unit circle, quadrants and graphs

Right-angled triangles only cover angles up to 90°. To handle bigger angles, and negative ones, trigonometry moves onto the unit circle, a circle of radius 1 centred on the origin. Draw a line from the centre at angle θ measured anticlockwise from the right-hand side. Where the line meets the circle, the point’s x-coordinate is cos θ and its y-coordinate is sin θ. Because the radius is 1, the triangle’s hypotenuse is 1 and the ratios turn into the coordinates directly.

Once you are going round the circle, the signs of the three ratios depend on the quadrant. A quick way to remember it is that sine is positive in the top half, cosine is positive on the right-hand side, and tangent is positive when sine and cosine have the same sign.

The reference angle is the acute angle between your line and the horizontal axis. Every angle shares its sine, cosine and tangent values with its reference angle, apart from the sign. For example 150° has a reference angle of 30°, so sin 150° = sin 30° = 0.5, while cos 150° = −cos 30° = −0.866. The Trig values tab works the reference angle and quadrant out for you.

Unroll the circle and you get the familiar wave. Here are sine and cosine plotted from 0° to 360°. The cosine curve is the sine curve shifted 90° to the left, and the two cross at 45° and 225°, where sine and cosine are equal. Waves like these describe sound, light and the alternating current in a UK wall socket, which is why they matter well beyond triangles.

Tangent is stranger. It climbs from 0 at 0°, heads for infinity as the angle approaches 90°, and then reappears from minus infinity on the other side. The dashed line at 90° is called an asymptote, and it is the reason tan 90° is undefined: tangent is sine divided by cosine, and cos 90° is 0.

Each step closer to 90° makes the tangent roughly ten times bigger. That is why the Calculator tab refuses to show a number for it. Any number it displayed would be wrong.

Exact trig values worth recognising

A handful of angles have neat exact values built from fractions and square roots. You will see these constantly in exam questions and in anything involving 30°, 45° and 60° set squares, so they are worth learning. Here is the main table, with the decimal values in the chart that follows it.

The Trig values tab goes further than this table. It gives the exact form for every multiple of 15°, so the less friendly angles are covered too.

These are the values behind the question “what is the exact value of tan 15 degrees?”. It is 2 − √3, about 0.2679. It comes from writing 15° as 45° − 30° and using the tangent subtraction formula.

Degrees, radians and gradians

There are three common ways to measure an angle, and a full turn is a different number in each. A degree is 1/360 of a turn. A radian is the angle you get when the arc along the circle is as long as the radius, so a full turn is 2π radians. A gradian splits a right angle into 100 equal parts, so a full turn is 400 gradians.

To switch between them, multiply degrees by π/180 to get radians, multiply radians by 180/π to get degrees, and multiply degrees by 10/9 to get gradians. One radian is about 57.2958°. As a quick degree to radian to gradian calculator reference, here are the angles you will meet most.

In practice you rarely need to convert by hand. On the Calculator tab, switch between DEG, RAD and GRAD and the same expression is interpreted in the new mode. On the Trig values tab, type an angle in degrees and the “In radians” card shows it as a fraction of π when it can.

Inverse trig: what does sin^-1 mean?

The little −1 in sin^-1 is the source of a lot of confusion. It does not mean “one over sine”. It means inverse sine, the function that undoes sine. Ask for the sine of 30° and you get 0.5. Ask for the inverse sine of 0.5 and you get 30° back. That is why what does sin inverse mean is one of the most searched questions about calculators. Many books write the same thing as arcsin, and the other two are arccos and arctan.

The ranges explain why a calculator’s inverse button gives only one answer. Plenty of angles share the same sine, so the calculator returns the one in its standard range, called the principal value. The Find angle tab then goes further and lists the others, plus the general solution. For tan θ = 1 you get 45° and 225°, with a general solution of 45° + 180°n, because tangent repeats every half turn.

Solving triangles: the sine rule and cosine rule

When a triangle has no right angle you cannot use SOH CAH TOA directly. Two rules take over. The law of sines says a / sin A = b / sin B = c / sin C, so it pairs each side with the angle opposite it. The law of cosines says a² = b² + c² − 2bc cos A, which is Pythagoras with a correction for the angle. Use the cosine rule when you know two sides and the angle between them (SAS), or all three sides (SSS). Use the sine rule when you know a side and its opposite angle plus one more piece.

Here are two examples you can check on the Triangle tab. First SAS. Sides b = 8 and c = 11 with the angle between them, A = 50°. The cosine rule gives a² = 64 + 121 − 176 cos 50° = 71.87, so a = 8.4776. The other angles are B = 46.2936° and C = 83.7064°. Then SSS. Sides 5, 7 and 9 give angles of 33.5573°, 50.7035° and 95.7392°, so this triangle is obtuse. The big angle comes from cos C = (25 + 49 − 81) / 70 = −0.1.

ASA and AAS are the friendliest cases. Knowing two angles gives you the third at once, because they add up to 180°, and then the sine rule fills in the sides. With c = 12, A = 40° and B = 65°, you find C = 75°, a = 7.9856 and b = 11.2593.

When SSA gives two triangles

SSA is the case where care is needed. Suppose you know side a, side b and angle A, with A acute. Work out b sin A, the height of the triangle if it were standing on side b. The number of triangles then depends on how long a is compared with that height.

In the example with a = 7, b = 10 and A = 40°, b sin A = 6.43. Since 7 is more than 6.43 but less than 10, two triangles fit. One has side c = 10.4322 and angles B = 66.6742°, C = 73.3258°. The other has c = 4.8887 and angles B = 113.3258°, C = 26.6742°. Many students lose marks by missing the second one, which is exactly why the calculator shows both.

Worked examples you can try right now

Reading about a tool only goes so far, so here are some everyday jobs with real numbers. Every one of them can be checked on the calculator at the top of this page.

A ladder against a wall

A 4 m ladder leans against a wall at 75° to the ground, close to the familiar 1 in 4 rule used on site, which works out at about 76°. How high does it reach, and how far from the wall is its foot? Choose Right triangle, enter side c = 4 and angle A = 75, with the unit set to m. The answer: it reaches a = 3.8637 m up the wall, and the foot sits b = 1.0353 m from the wall. By hand that is 4 sin 75° and 4 cos 75°.

The length of a roof rafter

A roof spans 6 m and has a 25° pitch. Each side runs 3 m horizontally. Enter side b = 3 and angle A = 25 in right triangle mode. The roof rises a = 1.3989 m, and the rafter along the slope is c = 3.3101 m, before any overhang. The gradient is tan 25° = 0.4663, or 46.6% as a slope percentage. If you want to play with the percentage side of things, our percentage calculator will do the arithmetic.

Height of a tree or building

Stand 20 m from the base of a tree. A clinometer app held at eye level, 1.6 m off the ground, reads 32° up to the top. The height above your eyes is 20 × tan 32° = 12.4974 m, so the tree is about 12.5 + 1.6 = 14.1 m tall. Type 20*tan(32) into the Calculator tab in DEG mode to check. In feet and inches that is roughly 46 ft 3 in.

Is that corner really square?

Builders have used the 3-4-5 trick for centuries. Measure 3 m along one wall and 4 m along the other. If the diagonal between the two marks is exactly 5 m, the corner is a true right angle. Enter 3, 4 and 5 as sides a, b and c in Any triangle mode and the calculator confirms the largest angle is 90°. Scale it up, for example 0.6, 0.8 and 1.0 for a small frame, and it works just the same.

Working from a scale drawing

Plans are usually drawn to scale, so a measurement taken off the paper is not the real length. Convert each length first, for instance with our scale calculator, and then enter the real lengths into the Triangle tab. Because similar triangles keep the same angles, the angles you get from the plan are already correct.

That table is what the Triangle tab returns when you enter sides 3 and 4 in right triangle mode. Notice that the circumradius of any right triangle is half the hypotenuse (2.5), a nice check that the numbers hang together.

Common mistakes and how to avoid them

• Wrong angle mode. sin(30) is 0.5 in degrees but about −0.988 in radians. Always check the DEG, RAD or GRAD badge before you start.
• Mixing units. Sides must all be in the same unit. Convert first, then enter.
• Labelling the triangle wrongly. Opposite and adjacent depend on which angle you are looking from. Side a is always opposite angle A.
• Reading sin^-1 as 1/sin. The reciprocal of sine is cosecant. The inverse sine is a different thing.
• Stopping at the first angle. In the sine rule an inverse sine can give an acute and an obtuse answer. Check whether both fit, especially in SSA.
• Rounding too early. Keep the full decimal in your working and round only the final answer. The calculator keeps up to 12 significant figures for you.
• Expecting angles alone to fix a size. Three angles give the shape of a triangle but not how big it is. You always need at least one side.

Trigonometry is only one part of a maths or science grade, so a single slip here rarely tells the whole story of a result.

Frequently asked questions

Tap a question to see the answer.

Add this tool to your website

Would you like this calculator on your own school, blog or maths help page? Copy the code below and paste it into the HTML of your page, in the place where you want the tool to appear.

Change the height number if you want a taller or shorter box. The calculator keeps working inside the frame exactly as it does here.