Arctan Calculator
What the Arctan Calculator finds
Ever wondered which angle hides behind a tangent value? Enter the ratio and the Arctan Calculator returns the matching angle. Engineers convert rise-over-run grades into incline angles, game programmers use it for atan2 style direction math, and students verify triangle solutions with it.
How the inverse works, with numbers
Arctan inverts the definition tan(x) = sin(x)/cos(x) on its principal branch. Because tan(45 degrees) = 1, arctan(1) = 45 degrees; because tan(60 degrees) = 1.732, arctan(1.732) = 60 degrees. A roof that rises 0.577 units for every unit across therefore has arctan(0.577) = 30 degrees.
Grades give the Tangent field a natural workout. A road signed as a 10 percent climb rises 10 units for every 100 of run, so enter 0.10, choose degrees, and press Calculate; the screen answers arctan(0.10) = 5.71 degrees. A gentler path at 8 percent works out to arctan(0.08) = 4.57 degrees, and a severe mountain pass at 20 percent reaches arctan(0.20) = 11.31 degrees. Notice the angle grows slower than the ratio itself, because tangent stretches steeply near vertical. In radians the 10 percent case is 0.0997, barely below the ratio, which is why small-grade shortcuts work in either unit. Arctan(0.05) rounds to 2.86 degrees for the flattest signed ramps.
Input rules, range, and units
Type a unitless ratio into the Tangent field, choose degrees or radians, and press Calculate. Any real number is acceptable, and answers always fall between -90 and 90 degrees, or -pi/2 and pi/2 radians. If your diagram needs the second quadrant angle, add 180 degrees to a negative result.
Frequently Asked Questions
What is arctan of 1?
Arctan(1) returns 45 degrees because tan(45 degrees) = 1 exactly, describing equal rise and run across the grid. In radians the answer is pi/4, approximately 0.7854.
What range can arctan return?
Answers fall between -90 and 90 degrees with endpoints excluded, since tangent is undefined at 90 degrees. So arctan(1) = 45 degrees while arctan(-1) = -45 degrees.
How do I convert slope to angle?
Divide rise by run to form the tangent, then take arctan. A rise of 0.577 over a run of 1 gives arctan(0.577) = 30 degrees, while 1.732 corresponds to 60 degrees.
What does a negative input mean?
A negative ratio returns a negative angle: arctan(-1.732) = -60 degrees. If the angle sits in the second quadrant of your diagram, add 180 degrees to get 120 degrees.
Is arctan the same as cot?
No. Cotangent is 1/tan(x), a length ratio; arctan is the inverse that recovers the angle. For instance arctan(0.577) = 30 degrees, while cot(30 degrees) is itself 1.732.