Calculate tangent for an angle in degrees and inspect its sine-over-cosine relationship.
Complete the required fields and select the calculate button.
Tangent is undefined wherever cosine is zero, including 90° plus any multiple of 180°.
The tangent calculator evaluates tan(α) for an angle entered in degrees. Tangent compares vertical change with horizontal change on the unit circle and, for an acute right-triangle angle, equals the opposite leg divided by the adjacent leg.
The result is dimensionless. Tangent repeats every 180 degrees and is undefined wherever cosine is zero, including 90°, 270°, and their coterminal angles.
Reduce a large angle by multiples of 180 degrees if you want an easy mental check, because tangent has period 180 degrees.
tan(α) = sin(α) ÷ cos(α)
For an acute angle in a right triangle, tan(α) = opposite ÷ adjacent. Degree conversion is radians = degrees × π/180. The quotient is defined only when cos(α) ≠ 0.
| Symbol or term | Meaning |
|---|---|
| α | Input angle in degrees. |
| Opposite | Leg across from the chosen acute angle. |
| Adjacent | Nearby leg other than the hypotenuse. |
| tan(α) | Dimensionless ratio, rounded to six decimals. |
| k | Any integer in repeating-angle descriptions. |
For α = 45°, sine and cosine are both √2/2. Their quotient is 1, so the page returns 1. In a right triangle this means the opposite and adjacent legs have equal length.
For 135°, tangent is also −1 because the angle is in quadrant II. At 90°, cosine is zero, so the page reports Undefined instead of displaying a large floating-point artifact.
A positive result occurs in quadrants I and III, where sine and cosine share a sign. A negative result occurs in quadrants II and IV. Tangent is zero at integer multiples of 180 degrees.
Tangent can have any real finite value. Its magnitude grows rapidly near 90° + 180°k because cosine approaches zero. A tiny change in angle there can create a large change in output, so measurement uncertainty matters.
For slope work, tan(α) gives rise divided by run when α is the directed angle from the positive horizontal axis. Multiplying by 100 converts that ratio to percent grade. A 45° angle corresponds to a 100% grade, not a 45% grade.
The reciprocal of tangent is cotangent, not inverse tangent. Arctangent starts with a ratio and returns a principal angle. The notation tan⁻¹ is sometimes used for arctangent and can therefore be confusing.
Right-triangle language applies directly to acute angles. The unit-circle definition extends tangent to negative angles and other quadrants, where a signed coordinate ratio describes direction.
Six displayed decimal places do not make an input exact. If an angle came from measurement, round the final use according to its uncertainty. Keep more precision for intermediate multiplication and round once at the end.
Useful checks are tan(0°) = 0, tan(45°) = 1, tan(−45°) = −1, and tan(α + 180°) = tan(α).
When finding a missing triangle side, rearrange the ratio carefully. If the adjacent side is known, opposite = adjacent × tan α. If the opposite side is known, adjacent = opposite ÷ tan α, provided tangent is nonzero. These relationships assume a right triangle and the chosen acute angle.
Signs carry geometric information outside an acute triangle. Do not discard a negative tangent unless the physical problem separately asks for an unsigned length or magnitude.
It uses degrees. Convert radians to degrees by multiplying by 180/π before entering the angle.
Tangent is sine divided by cosine. At 90 degrees cosine is zero, and division by zero has no finite real value.
Adding 180 degrees changes both sine and cosine signs. Their quotient therefore stays the same wherever it is defined.
It equals slope when the angle is measured from the positive horizontal direction and the axes use compatible scales. Context and direction still matter.
Multiply the tangent ratio by 100. For example, tangent of 45 degrees is 1, which corresponds to a 100% grade.
Arctangent is the inverse function that returns a principal angle from a ratio. Cotangent is the reciprocal, equal to one divided by tangent.
Check whether the other tool uses radians, degrees, or gradians, and compare rounding. Near undefined angles, very small input differences have a large effect.