Enter one angle in degrees or radians and get sine, cosine, tangent, cotangent, secant, and cosecant all at once — plus the quadrant, with undefined ratios clearly labeled.
Enter an angle to see all six trig ratios
This trigonometry calculator is a hub tool: instead of computing one ratio at a time, it takes a single angle — in degrees or radians — and instantly returns all six trigonometric ratios together: sine, cosine, tangent, cotangent, secant, and cosecant. It also shows which of the four quadrants the angle falls in. Where a ratio is mathematically undefined at that exact angle (like tangent at 90°), the calculator explicitly labels it "Undefined" instead of showing an error, "Infinity," or a distractingly huge number — so you always get an honest, complete snapshot of an angle's full trigonometric profile in one step.
Every angle has six associated ratios built from the unit circle's x- and y-coordinates (cosine and sine) and their combinations. This calculator converts your angle to radians internally, evaluates sin and cos directly, then derives tan, cot, sec, and csc from those two base values — checking for near-zero denominators at each step so asymptotes are reported correctly rather than silently breaking.
This tool suits trigonometry and precalculus students who need to see all six ratios of an angle side by side for homework or exam review, teachers building worked examples, and anyone in physics, engineering, or navigation who works with secant, cosecant, or cotangent alongside the more familiar sine, cosine, and tangent and wants them computed together rather than one at a time on a scientific calculator.
The six ratios describe how an angle relates the sides of a right triangle or the coordinates of a point on the unit circle. Because three of them (cot, sec, csc) are simply reciprocals of the other three, understanding all six together — including exactly where they stop being defined — builds a much more complete picture of how trigonometric functions behave than learning sine, cosine, and tangent in isolation.
Physicists resolve forces and velocities into components using sine and cosine, then use tangent to recover an angle. Electrical engineers use phase angles and their trig ratios in AC circuit analysis. Surveyors and navigators combine multiple ratios of the same bearing angle. Computer graphics and game engines use all six ratios together when rotating vectors or building rotation matrices.
How sin and cos generate the other four ratios
sin/csc, cos/sec, and tan/cot are each mathematical reciprocals — whenever one member of a pair is zero, the other is undefined.
Every angle has exactly six ratios, but only two (sine and cosine) are computed directly — the rest are algebraic combinations of those two.
Tan and sec break down at 90°/270°; cot and csc break down at 0°/180°/360° — this calculator flags those cases instead of showing a huge number.
From entering an angle to reading all six ratios
Type the angle value into the input field — any real number, positive or negative, is accepted.
Select which unit your angle is measured in using the toggle above the input.
The calculator converts your angle to radians internally and evaluates all six ratios.
Review sin, cos, tan, cot, sec, and csc together — any ratio that's mathematically undefined at that angle is clearly labeled instead of showing an error.
See which of the four quadrants (or which axis) the angle falls on, shown as a bonus result below the six ratios.
Finding all six ratios of a 30° angle
A student needs sin, cos, tan, cot, sec, and csc for a 30° angle to complete a trigonometry worksheet.
Undefined case: at 90°, cos 90° ≈ 0, so tan and sec are both Undefined — while sin = 1, cos = 0, cot = 0, and csc = 1 remain perfectly valid. The calculator shows exactly this mix of defined and undefined ratios rather than crashing or displaying "Infinity."
Domain, range, and period of each of the six ratios
| Ratio | Range | Period | Undefined At |
|---|---|---|---|
| sin θ | [-1, 1] | 360° | Never |
| cos θ | [-1, 1] | 360° | Never |
| tan θ | All real numbers | 180° | 90°, 270°, ... |
| cot θ | All real numbers | 180° | 0°, 180°, 360°, ... |
| sec θ | (-∞,-1] ∪ [1,∞) | 360° | 90°, 270°, ... |
| csc θ | (-∞,-1] ∪ [1,∞) | 360° | 0°, 180°, 360°, ... |
Reading the results together: sine and cosine always stay between -1 and 1, so their reciprocals (csc and sec) can never land between -1 and 1 — they jump straight from 1 to infinity and back. Tangent and cotangent, by contrast, can be any real number at all.
Manual verification: once you have sin and cos from a calculator or reference table, you can hand-check tan, cot, sec, and csc yourself with simple division — tan = sin ÷ cos, cot = cos ÷ sin, sec = 1 ÷ cos, csc = 1 ÷ sin.
Quadrant signs: Quadrant I has all six ratios positive; Quadrant II only sin and csc positive; Quadrant III only tan and cot positive; Quadrant IV only cos and sec positive — a quick way to sanity-check a result's sign.
Where seeing all six ratios of one angle together is genuinely useful
Check every ratio of an angle at once instead of computing them one at a time.
Build lesson examples that show how all six ratios relate to a single angle.
Break a force or velocity into components and check the related ratios together.
Evaluate AC circuit phase relationships that use secant or cosecant alongside sine.
Compute multiple ratios of the same bearing or elevation angle in one pass.
Build rotation matrices or camera angles that need sin, cos, and tan together.
Solve joint angles where multiple trig ratios of the same angle appear in one equation.
Relate a phase angle's sine and cosine components in one combined view.
Evaluate multiple trigonometric ratios for a given orbital or elevation angle.
Check roof pitch, truss, or support angles across every relevant ratio.
Quickly verify hand-worked trigonometry answers before a test or quiz.
Get an instant six-ratio reference table for any angle without a textbook.
Confirm sin×csc=1, cos×sec=1, and tan×cot=1 hold for your specific angle.
What this all-in-one trig calculator does well, and where it doesn't apply
Sin, cos, tan, cot, sec, and csc side by side at 0°, 30°, 45°, 60°, and 90°
| Angle | sin | cos | tan | cot | sec | csc |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | Undefined | 1 | Undefined |
| 30° | 0.5 | 0.866025 | 0.577350 | 1.732051 | 1.154701 | 2 |
| 45° | 0.707107 | 0.707107 | 1 | 1 | 1.414214 | 1.414214 |
| 60° | 0.866025 | 0.5 | 1.732051 | 0.577350 | 2 | 1.154701 |
| 90° | 1 | 0 | Undefined | 0 | Undefined | 1 |
Summary: This trigonometry calculator gives you an instant, free way to see all six ratios of a single angle at once, with undefined values clearly flagged. Pair it with the Unit Circle Calculator and Triangle Calculator for a fuller trigonometry toolkit.
Common questions about the six trigonometric ratios
Trusted educational references to go deeper on trigonometric ratios
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