What is tangent
The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side: tan(θ) = opposite ÷ adjacent = sin(θ) ÷ cos(θ). For example, tan(45°) = 1 because the two legs are equal.
Unlike sine and cosine, tangent has no upper or lower limit. Both legs of the triangle can vary freely, so as the angle approaches 90° the opposite side grows without bound against a shrinking adjacent side and the ratio races off towards infinity. Because tangent is rise over run, it is the same quantity handled by the slope calculator for a straight line.
How to use this calculator
Enter the angle in degrees and read tan(θ) rounded to four decimals. To go from a slope value back to the angle, use the arctangent calculator.
Make sure your angle really is in degrees: entering a radian value such as 0.7854 returns the tangent of 0.7854 degrees, about 0.0137, rather than the 1 you would get from 45°. Useful checkpoints are tan(0°) = 0, tan(45°) = 1 and tan(60°) ≈ 1.7321. Negative angles give negative tangents, since tan(−θ) = −tan(θ).
Where tangent is undefined
Tangent equals sine divided by cosine, so it is undefined wherever cosine is zero — at 90°, 270° and every 180° beyond. The calculator flags these cases instead of returning a value.
Approaching those angles the output becomes enormous and highly sensitive: tan(89°) is about 57, while tan(89.9°) is close to 573. Treat very large tangent values with caution, because a tiny measurement error in the angle produces a huge swing in the result. Tangent also repeats every 180° rather than every 360°, so tan(225°) equals tan(45°).
Frequently asked questions
- What is tan(45°)?
- tan(45°) = 1, because the opposite and adjacent sides of a 45-degree right triangle are equal.
- Why is tan(90°) undefined?
- Tangent is sine ÷ cosine, and cos(90°) = 0, so dividing by zero leaves it undefined.
- Does this calculator use degrees?
- Yes, enter the angle in degrees; it is converted to radians before computing.
- What is the formula for tangent?
- tan(θ) = opposite ÷ adjacent, equivalently sin(θ) ÷ cos(θ).
- How is tangent related to the slope of a line?
- The tangent of a line's angle of inclination is exactly its slope, since slope is rise over run and tangent is opposite over adjacent. A 45° line has slope 1, and a vertical line has an undefined slope for the same reason tan(90°) is undefined.