Tan Angle Calculator: Free Trigonometry Tool
Tangent is the workhorse ratio of trigonometry: opposite over adjacent, rise over run, the steepness of an angle turned into a single number. This tan angle calculator runs it in four directions. Get tan(θ) from any angle, run arctan to turn a tangent value back into an angle, convert a rise-over-run slope into degrees, or compute all six trig functions at once, in degrees, radians or gradians.
| Calculation mode | |
| Input value | |
| Angle in degrees | |
| Angle in radians | |
| Angle in gradians | |
| tan(θ) | |
| sin(θ) | |
| cos(θ) | |
| cot(θ) = 1/tan | |
| sec(θ) = 1/cos | |
| csc(θ) = 1/sin | |
| Pythagorean identity check | |
| tan identity check | |
| Reference angle |
| Angle (°) | Radians | tan(θ) | sin(θ) | cos(θ) | cot(θ) |
|---|
Results use standard mathematical trigonometric formulas, accurate to 10 decimal places. tan(θ) is undefined at 90°, 270° and equivalent angles. For critical calculations, verify your inputs before relying on the results.
Quick Answer
tan(45°) = 1 exactly, with sin and cos both 0.7071067812. Run the inverse and arctan(1) = 45°. A rise of 3 over a run of 4 is a 36.8698976458° slope, which is a 75% grade. The calculator opens with all three values pre-filled, so one click per mode reproduces these numbers.
Key takeaways
- tan(θ) = opposite ÷ adjacent = sin(θ) ÷ cos(θ). It is the only trig function with no fixed range: any real number is possible.
- arctan runs the question backwards and always returns the principal angle between -90° and 90°. A dedicated arctan calculator covers that direction in depth, including the quadrant-aware atan2 case.
- Slope mode turns rise over run into an angle plus the grade percentage, the form engineers and builders actually quote.
- tan is undefined at 90° and 270° (and every angle coterminal with them) because cos(θ) is zero there. This tool says "Undefined" instead of printing a meaningless giant number.
How to use the tan angle calculator
Pick a mode, enter the number, click Calculate. The results stay hidden until you calculate.
- Choose the mode: tan(θ) for the forward tangent, arctan / tan⁻¹ for the inverse, Slope → Angle for rise-over-run, or All Trig Functions for the full six-function breakdown. The mode tabs sit at the top of the card.
- Enter your value: an angle for tan and all-functions modes, any real number for arctan, a rise and a run for slope mode. The worked values are pre-filled: 45 for tan, 1 for arctan, 3 and 4 for slope, 60 for all-functions.
- Pick the unit: degrees, radians or gradians. This matters enormously: tan(45) in degrees is 1, in radians it is 1.6197751905. The most common trig mistake is a right calculation in the wrong unit.
- Click Calculate: the hero result appears with sin and cos cards, a quadrant badge, a full breakdown of all six functions, the identity checks, and the 25-angle reference table. Copy or share the result from the buttons in the results area.
Invalid inputs are rejected with an inline message rather than a guess: a blank or non-numeric angle ("Please enter a valid angle value."), a blank tangent value ("Please enter a valid tangent value."), a blank rise or run, and the one impossible slope, rise = 0 with run = 0 ("Both rise and run cannot be zero.").
What each mode computes
Four modes, one shared engine. Here is the math behind each.
tan(θ) mode evaluates the tangent of your angle. In a right triangle, tan(θ) = opposite ÷ adjacent; on the unit circle it is sin(θ) ÷ cos(θ), the slope of the line from the origin to the point at angle θ. The result prints to 10 decimal places.
arctan mode answers the reverse question: which angle has this tangent? Enter 1 and it returns 45°, enter -0.577 and it returns about -29.98°. The answer is always the principal value between -90° and 90°. Angles outside that range share their tangent with an angle inside it (tan repeats every 180°), so if your problem needs the full circle, the quadrant has to come from extra information, which is what atan2(y, x) is for.
Slope → Angle mode computes θ = arctan(rise ÷ run) and also reports the grade percentage, rise ÷ run × 100. A run of zero means a vertical slope, reported honestly as 90° with an infinite grade rather than a crash. This is the mode surveyors, roofers and road engineers reach for: a 3-4-5 triangle gives 36.8698976458° and a 75.0000% grade.
All Trig Functions mode evaluates tan, sin, cos, cot, sec and csc for one angle, with the unit-circle position, the reference angle, and both identity checks. The cosine calculator and the reference angle calculator cover the neighboring questions if you need those topics on their own.
Units: degrees divide the circle into 360, radians into 2π (a radian is the angle whose arc equals the radius), gradians into 400. Conversions are exact: radians = degrees × π ÷ 180, gradians = degrees × 200 ÷ 180. The tool converts your input to radians internally, computes, then converts the answer back.
Worked examples: tan(45°) and a 3-4 slope
Two runs show the tool at full stretch. First, tan mode with 45°, which is the cleanest angle in trigonometry:
| Function | Value at 45° | Why |
|---|---|---|
| tan(45°) | 1 | sin and cos are equal, so their ratio is 1 |
| sin(45°), cos(45°) | 0.7071067812 | √2 ÷ 2 |
| cot, sec, csc | 1, 1.4142135624, 1.4142135624 | reciprocals of tan, cos, sin |
| In radians / gradians | 0.7853981634 rad / 50 grad | π/4 rad; a right angle is 100 grad |
Second, slope mode with rise 3 and run 4: the angle is arctan(0.75) = 36.8698976458°, the grade is 75.0000%, and sin/cos come out as the tidy 0.6 and 0.8 of the 3-4-5 triangle. Five more real runs of the tool, reproducible by typing the values in:
| Mode and input | Result |
|---|---|
| tan(30°) | 0.5773502692 |
| tan(60°) | 1.7320508076 |
| tan(90°) | Undefined |
| arctan(-0.577) in radians | -0.5233360339 rad |
| Slope rise 1, run 0 (vertical) | 90°, grade ∞ (vertical) |
Reading your results: quadrant, reference angle, and the checks
Every result carries three layers of self-verification, which is what separates a careful tool from a bare function call.
- Quadrant badge. Your angle is normalized to 0°–360° and placed: Quadrant I through IV, or on an axis (0°, 90°, 180°, 270°). Tangent is positive in Quadrants I and III and negative in II and IV, so the badge is a quick sign check on the answer.
- Reference angle. The acute angle between your angle's terminal side and the x-axis. tan(150°) has reference angle 30°, and indeed tan(150°) = -tan(30°) = -0.5773502692. The sign comes from the quadrant, the magnitude from the reference angle.
- Identity checks. The Pythagorean check confirms sin²(θ) + cos²(θ) = 1, and the tan check shows the actual division sin ÷ cos landing on the reported tangent. If both print clean, the three primary values are internally consistent.
One honest edge case: feed the tool an angle in radians that lands a hair past 90°, like 1.5707963268, and the hero correctly reports Undefined while the quadrant badge reads Quadrant II. That is floating-point truth, not a bug: the angle genuinely is 90.0000000001°, past the axis, while the undefined-detection threshold caught the pole. The hero answer is the one to trust.
Below the breakdown sits the 25-angle reference table, every 15° from 0° to 360°, with tan, sin, cos and cot. Poles print as "Undef." rather than as garbage digits.
Where other tan and arctan tools fall short
Most tangent tools do exactly one job. This page's bet is that the four jobs belong together, because students and engineers keep converting between them:
| Tool | What it does | What it leaves out |
|---|---|---|
| This calculator | tan, arctan, slope→angle, and all six functions; degrees, radians, gradians | No atan2 two-argument mode; arctan stays on the principal range (stated in the math section) |
| CalculatorX tan calculator | Forward tan with exact common-angle labels and explicit pole classification | Forward only: no arctan, no slope mode, no six-function breakdown |
| NeftCal tan calculator | tan plus cot, with undefined cases clearly labeled | Tan only, degrees and radians only, no slope or inverse |
| CalculatorLib tangent calculator | A simple angle-in, tangent-out box | Thin: no inverse, no slope, no verification of any kind |
| EasyUnitConverter arctan | Inverse tangent with a quadrant-aware atan2(y, x) mode | Inverse only; atan2 is genuinely useful and this page does not offer it |
| EpicCalculators sin-cos-tan | All three primary functions plus reciprocals and unit-circle position | No slope mode, no gradians, no reference table |
The gap this page fills is the slope direction: rise-over-run to angle is one of the most-searched tangent tasks in construction and civil work, and none of the single-purpose tools above does it. The identity checks and the reference table are the second gap: they turn an answer into an answer you can trust.
Frequently asked questions
Tan runs forward: angle in, ratio out (tan(45°) = 1). Arctan runs backward: ratio in, angle out (arctan(1) = 45°). They undo each other inside arctan's principal range of -90° to 90°.
Because tan(θ) = sin(θ) ÷ cos(θ), and cos(90°) = 0. Division by zero has no value, so the tangent blows up to positive infinity from one side and negative infinity from the other. The calculator reports "Undefined" instead of a misleading huge number.
Gradians divide a right angle into 100, so a full circle is 400 grad. Surveyors and some European engineering fields use them because a right angle being exactly 100 is convenient. 90° = 100 grad, 45° = 50 grad.
Divide rise by run and take the arctangent: angle = arctan(rise ÷ run). A rise of 3 over a run of 4 gives arctan(0.75) = 36.8698976458°, a 75% grade. Use the Slope → Angle mode and it also reports the grade percentage.
Where your normalized angle sits on the circle. It is a sign check: tangent is positive in Quadrants I and III and negative in Quadrants II and IV. If the badge disagrees with your expectation, check the angle or the unit.
The acute angle between your angle's terminal side and the x-axis. It carries the magnitude of the trig values while the quadrant carries the sign: tan(150°) = -tan(30°) because 150° sits in Quadrant II with reference angle 30°.
Mathematically, tan(90°) is undefined. A plain calculator evaluates the nearest floating-point number to 90°, which is not exactly 90°, and returns an enormous finite value. That number is an artifact of rounding, not the tangent of 90°. This tool detects the pole and says Undefined.
Yes. It is free, runs entirely in your browser, and asks for no account, so your angle inputs never leave your device. “Copy my results” grabs the tan or arctan result together with the quadrant badge, reference angle and grade percentage in one go; “Share results” sends the same summary wherever you need it.
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References
- Eric W. Weisstein. Trigonometric Functions, MathWorld (definitions of the six circular functions, tan = sin/cos, periodicity, SOHCAHTOA). Accessed October 24, 2026.
- Wolfram Research. Tan, Wolfram Language Documentation (tangent defined as the ratio of sine and cosine; argument assumed in radians; exact values at rational multiples of π). Accessed October 24, 2026.
- Wolfram Research. TanDegrees: Calculate the tangent function in degrees, Wolfram Language Documentation (degree-based tangent, sum and half-angle identities, solving trig equations). Accessed October 24, 2026.
- Paul Dawkins. Algebra Trig Review: Trig Function Evaluation, Paul's Online Math Notes, Lamar University (degrees vs radians, unit circle evaluation of trig functions). Accessed October 24, 2026.
- Khan Academy. Inverse trigonometric functions (inverse trig as the reverse question: angle from ratio, calculator use in degree mode). Accessed October 24, 2026.