Tangent Calculator
Calculate the tangent of any angle in degrees instantly. Use it when finding the slope of a line, solving right-triangle problems, or determining the ratio of rise to run in construction and navigation.
About this calculator
The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle: tan(θ) = opposite / adjacent. It can also be expressed as the ratio of sine to cosine: tan(θ) = sin(θ) / cos(θ). In degree-based calculations the formula applied is tan(θ) = tan(angle × π / 180). Unlike sine and cosine, tangent is not bounded — its values range from −∞ to +∞. It equals zero at 0°, equals 1 at 45°, and approaches infinity as the angle nears 90° (where cosine approaches zero). Tangent is especially useful in surveying for measuring heights from a known horizontal distance, in engineering for calculating slopes and grades, and in physics for resolving force directions.
How to use
Suppose you want to find the tangent of 30°. Enter 30 in the Angle field. The calculator converts it: 30 × π / 180 ≈ 0.5236 radians. It then computes tan(0.5236) ≈ 0.5774. This means for every 1 unit of horizontal distance, the vertical rise is about 0.5774 units at a 30° incline. Practically, if a ramp extends 10 m horizontally at a 30° angle, the vertical height is 10 × 0.5774 ≈ 5.77 m.
Frequently asked questions
What does the tangent of an angle mean in practical terms?
Tangent represents the slope — the ratio of vertical rise to horizontal run — at a given angle. A tangent of 1 means a 45° incline where rise equals run. A tangent of 0.5 means the surface rises 0.5 units for every 1 unit traveled horizontally. This makes tangent the go-to function in construction, road design, and surveying when you need to relate an angle to a physical slope or gradient.
Why is the tangent undefined at 90 degrees?
At 90°, the adjacent side of the right triangle shrinks to zero, and dividing by zero is mathematically undefined. As the angle approaches 90° from below, the tangent grows without bound toward positive infinity. On the unit circle, this corresponds to the cosine (x-coordinate) approaching zero in the denominator of tan(θ) = sin(θ)/cos(θ). Because of this, tangent has vertical asymptotes at 90°, 270°, and every odd multiple of 90°.
How do I use the tangent function to find the height of a tall object?
If you know the horizontal distance to an object and the angle of elevation to its top, you can find the height using the formula: height = distance × tan(angle). For example, if you stand 50 m from a tree and measure an elevation angle of 35°, the height is 50 × tan(35°) ≈ 50 × 0.7002 ≈ 35 m. This technique is used in surveying, forestry, and navigation. Make sure your angle is measured from the horizontal, not the vertical, for the formula to apply correctly.