What the six trigonometric functions measure
Draw a circle of radius 1 centred on the origin, start at the point (1, 0), and rotate counter-clockwise through an angle θ. Wherever you land, the x-coordinate of that point is cos θ and the y-coordinate is sin θ. That is the whole definition. Everything else in trigonometry is a consequence of it.
The right-triangle definitions you may have learned first — sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent — are the same statement for acute angles. On the unit circle the hypotenuse is 1, so dividing by it changes nothing and the ratios collapse to the coordinates. The advantage of the circle definition is that it keeps working past 90°, where there is no right triangle to speak of, and it explains the signs: in the second quadrant x is negative and y is positive, so cosine is negative and sine is positive. ISO 80000-2, the international standard for mathematical signs and symbols, defines the six circular functions in exactly these terms, and its names — sin, cos, tan, cosec or csc, sec and cot — are the ones used on this page.
Tangent is the ratio y/x, which geometrically is the slope of the line from the origin through that point. It follows immediately that tangent is undefined wherever x = 0, at 90° and 270°, because a vertical line has no slope. The three reciprocal functions are exactly what their names say: cosecant is 1/sin, secant is 1/cos, and cotangent is cos/sin. They are undefined wherever their denominator vanishes, which is why csc and cot fail at 0° and 180° while sec and tan fail at 90° and 270°.
Signs, quadrants and periodicity
The sign of each function depends only on the signs of x and y, so it depends only on the quadrant. In quadrant I both coordinates are positive and all six functions are positive. In quadrant II only y is positive, so only sine and its reciprocal cosecant are positive. In quadrant III both coordinates are negative, so their ratio is positive and tangent and cotangent survive. In quadrant IV only x is positive, so cosine and secant are positive. The traditional mnemonic All Students Take Calculus names the surviving function in each quadrant in order.
Periodicity comes from the fact that a full revolution returns you to the same point. Sine, cosine, cosecant and secant all repeat every 360°, while tangent and cotangent repeat every 180°, because diametrically opposite points give the same slope. That is why an angle of 750° gives identical values to 30°: subtract two full turns and nothing changes. The coterminal angle calculator performs that reduction on its own.
Symmetry gives you two more free facts. Cosine is an even function, cos(−θ) = cos θ, because reflecting across the x-axis leaves x alone. Sine is odd, sin(−θ) = −sin θ, because the reflection flips y. Tangent, being their ratio, is odd too. Combined with the reference angle — the acute angle to the nearest part of the x-axis — these rules let you evaluate any angle from the first-quadrant table alone.
Worked example: evaluating all six functions at 240°
Take θ = 240° and work it out by hand before checking against the calculator.
- Locate the quadrant. 240° is between 180° and 270°, so it lies in quadrant III, where x and y are both negative and only tangent and cotangent are positive.
- Find the reference angle. Measured back to the negative x-axis, 240° − 180° = 60°. Every function's magnitude at 240° equals its magnitude at 60°.
- Look up the 60° values. sin 60° = √3/2 = 0.8660254, cos 60° = 1/2, tan 60° = √3 = 1.7320508.
- Apply the quadrant signs. sin 240° = −0.8660254; cos 240° = −0.5; tan 240° = (−0.8660254) ÷ (−0.5) = +1.7320508.
- Take reciprocals. csc 240° = 1 ÷ (−0.8660254) = −1.1547005; sec 240° = 1 ÷ (−0.5) = −2; cot 240° = 1 ÷ 1.7320508 = 0.5773503.
- Check with the Pythagorean identity. sin² + cos² = 0.75 + 0.25 = 1. The values are consistent.
That last check is worth doing on any hand computation. Because sin² θ + cos² θ = 1 is the equation of the unit circle itself, it holds for every angle without exception, and it catches a dropped sign or a mistyped surd immediately.
Reading the output, including the undefined cases
Sine and cosine always land between −1 and 1, so any value outside that band means the angle unit is wrong. That is the single most common failure: entering 30 while the calculator is in radian mode gives sin = −0.988, a perfectly valid number for a rotation of 30 radians and completely wrong for 30 degrees.
Tangent and cotangent are unbounded, so large magnitudes are legitimate. tan 89° = 57.29 and tan 89.9° = 572.96; the function grows without limit as the angle approaches 90°. At exactly 90° it does not have a large value, it has no value at all, and this calculator reports a dash rather than the enormous number a floating-point library would produce from a cosine of about 6 × 10⁻¹⁷.
Cosecant and secant are never strictly between −1 and 1, because they are reciprocals of quantities that never exceed 1 in magnitude. If you see a secant of 0.8, something upstream is wrong. Their minimum magnitude is exactly 1, reached where the underlying sine or cosine hits its extreme, and they grow without bound near their asymptotes exactly as tangent does.
Exact values at the standard angles
| Angle | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 120° | 2π/3 | √3/2 | −1/2 | −√3 |
| 135° | 3π/4 | √2/2 | −√2/2 | −1 |
| 150° | 5π/6 | 1/2 | −√3/2 | −√3/3 |
| 180° | π | 0 | −1 | 0 |
| 210° | 7π/6 | −1/2 | −√3/2 | √3/3 |
| 225° | 5π/4 | −√2/2 | −√2/2 | 1 |
| 240° | 4π/3 | −√3/2 | −1/2 | √3 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 300° | 5π/3 | −√3/2 | 1/2 | −√3 |
| 315° | 7π/4 | −√2/2 | √2/2 | −1 |
| 330° | 11π/6 | −1/2 | √3/2 | −√3/3 |
To get csc, sec and cot, take the reciprocal of the sin, cos and tan entry in the same row. A reciprocal of 0 is undefined, which is where the asymptotes live.
Common mistakes when evaluating trig functions
- Wrong angle mode. sin(30) is 0.5 in degrees and −0.988 in radians. Check the unit selector before trusting any answer.
- Confusing sin² θ with sin(θ²). The first squares the result, the second squares the angle. Only the first appears in the Pythagorean identity.
- Treating a huge tangent as an error. tan 89.99° = 5729.6 is correct; the function genuinely diverges. Only at exactly 90° is it undefined.
- Assuming csc is the same as sin⁻¹. Cosecant is the reciprocal 1/sin; sin⁻¹ is the inverse function arcsine. They are unrelated, and the inverse trig calculator handles the second.
- Rounding a surd too early. Writing √3/2 as 0.87 instead of 0.8660254 puts a 0.46% error into everything downstream of it, since 0.0039746 ÷ 0.8660254 = 0.0046.
Why 90° shows a dash and not a very large number
In double-precision arithmetic, cos(90°) does not evaluate to zero. Converting 90 to radians gives a value that is not exactly π/2 — π/2 is irrational and cannot be represented — so the cosine comes out around 6.1 × 10⁻¹⁷ and the tangent around 1.6 × 10¹⁶. That number is a floating-point artefact, not mathematics. This calculator snaps sines and cosines smaller than 10⁻¹² to exact zero and then reports the dependent function as undefined, which is what the mathematics actually says.
Where these values get used
Any right-triangle problem with one known angle and one known side reduces to a single multiplication by one of these values. A rafter on a 30° roof with a 12-foot horizontal run has length 12 ÷ cos 30° = 12 ÷ 0.8660254 = 13.856 feet, and rises 12 × tan 30° = 6.928 feet. The right triangle calculator and the Pythagorean theorem calculator automate the rest of that arithmetic.
Oblique triangles need the same values inside the law of sines and the law of cosines, which are what you reach for when there is no right angle to exploit. Vector resolution uses them too: a 500-newton force at 40° above horizontal has a horizontal component 500 cos 40° = 383.0 N and a vertical component 500 sin 40° = 321.4 N.
Beyond geometry, sine and cosine describe anything that oscillates: alternating current, sound pressure, the tide, a pendulum, a spring. A signal written as A sin(2πft + φ) uses the same function evaluated at an angle that grows linearly with time, which is why the argument is normally kept in radians and converted with the degrees to radians calculator. When you need to run the process backwards — from a ratio to an angle — use the inverse trig functions calculator, and when you need the acute equivalent of a large rotation, the reference angle calculator is the shortcut.