Calculate the tangent ratio (tan θ = opposite/adjacent) for any angle or triangle sides. Find missing values, view SOH-CAH-TOA reference, common tangent values table, and side ratio visualization.
The tangent ratio is one of the three fundamental trigonometric ratios, defined for an acute angle θ in a right triangle as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle: tan(θ) = opposite / adjacent. It is the "TOA" part of the classic mnemonic SOH-CAH-TOA that students learn alongside sine and cosine.
Tangent is ubiquitous in real-world applications. Surveyors use it to calculate heights of buildings from a known distance and angle of elevation. Engineers use it to determine slopes, grades, and inclines. Navigation, physics, and architecture all rely on tangent calculations. Even the grade of a road (rise over run) is a tangent ratio.
This calculator works in two directions. Given an angle (in degrees or radians), it computes the tangent value along with the corresponding sine and cosine. Given the opposite and adjacent sides of a right triangle, it computes the angle (via arctangent), the tangent ratio, the hypotenuse, and a visual breakdown of side proportions. A reference table of common tangent values for standard angles (0° through 360°) is always visible for quick lookup, and the SOH-CAH-TOA mnemonic table provides a handy reminder of all three basic ratios.
Use the presets for standard angles and common Pythagorean triples, or enter your own values. The calculator handles any real angle, not just those in the first quadrant.
Tangent Ratio Calculator helps you solve tangent ratio problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Decimal places, Angle, Opposite side once and immediately inspect tan(θ), Angle (degrees), Angle (radians) to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
tan(θ) = opposite / adjacent. Inverse: θ = arctan(opposite / adjacent). Related: sin(θ) = opposite / hypotenuse, cos(θ) = adjacent / hypotenuse, hypotenuse = √(opposite² + adjacent²).
Result: tan(θ) shown by the calculator
Using the preset "30°", the calculator evaluates the tangent ratio setup, applies the selected algebra rules, and reports tan(θ) with supporting checks so you can verify each transformation.
This calculator takes Decimal places, Angle, Opposite side, Adjacent side and applies the relevant tangent ratio relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use tan(θ), Angle (degrees), Angle (radians), sin(θ) to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
The tangent ratio is the ratio of the opposite side to the adjacent side in a right triangle: tan(θ) = opposite / adjacent. Use this as a practical reminder before finalizing the result.
Tangent is undefined when cos(θ) = 0, which happens at 90°, 270°, and their coterminal angles (90° + 180°n). At these points, the adjacent side has zero length.
SOH = Sine is Opposite over Hypotenuse. CAH = Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent.
Yes. The tangent function is defined for all angles (except where undefined). For obtuse angles, tangent is negative in the second quadrant and positive in the third quadrant.
The tangent of the angle a line makes with the horizontal axis equals the line's slope. A 45° angle gives a slope of 1; a steeper angle gives a larger tangent value.
tan(30°) = √3/3 ≈ 0.5774 and tan(60°) = √3 ≈ 1.7321. These come from the special 30-60-90 triangle with side ratios 1 : √3 : 2.