Tangent Calculator
What the Tangent Calculator computes
A Tangent Calculator converts an angle into the tangent ratio, defined as tan(x) = sin(x)/cos(x). Civil engineers use it for road grades, surveyors for elevation angles, and students for confirming hand calculations before an exam.
The sine over cosine method
The formula divides the sine by the cosine of the same angle. At 45 degrees, sin(45) = 0.7071 and cos(45) = 0.7071, so tan(45) = 0.7071/0.7071 = 1. At 60 degrees, sin(60) = 0.8660 while cos(60) = 0.5, giving tan(60) = 0.8660/0.5 = 1.732, the classic value of the 30-60-90 triangle.
Using the Angle field correctly
Enter the angle in the Angle box, pick degrees or radians, then press Calculate. Every real angle is valid except 90 degrees plus multiples of 180, where cosine equals zero and tangent is undefined. The range of results spans all real numbers, and negative angles simply produce negative tangents.
A site example shows the tool at work. Suppose a surveyor reads a 30 degree elevation angle between the instrument line and the ground; typing 30 into the Angle box, keeping the unit on degrees, and pressing Calculate returns tan(30 degrees) = 0.5774. The line therefore climbs about 0.58 meters for every meter forward, a slope crews can verify with a level and a tape. The identical angle entered as 0.5236 radians yields 0.5774 as well, which is why matching the unit selector to how the angle was measured matters before every run. Checking a second point, 15 degrees produces 0.2679, so halving the angle cuts the ratio far faster than half.
Frequently Asked Questions
Why does tan 45 equal 1?
At 45 degrees, sine and cosine are equal at about 0.7071 each, and 0.7071 divided by 0.7071 is exactly 1. That is also why a 45 degree incline has a 100 percent grade.
What is tan 60 degrees?
Tan(60 degrees) equals 1.732 because sin(60) = 0.8660 and cos(60) = 0.5, so 0.8660/0.5 = 1.732. Engineers quote this constant for steep 60 degree chutes and ramps.
Why is tan 90 undefined?
Cosine equals zero at 90 degrees, and tan(90) would mean dividing sin(90) = 1 by that zero. The tangent curve therefore rises without bound, so no finite value exists there.
Can tangent results be negative?
Yes. Tangent repeats every 180 degrees and is negative in the second and fourth quadrants, so tan(120 degrees) = -1.732 and tan(-45 degrees) = -1.
How do radians change the input?
Radians measure angles differently: 45 degrees equals 0.7854 radians and 60 degrees equals 1.0472 radians. Enter the number matching your chosen unit, since tan(1.0472) still returns 1.732.