How this instrument works
Tangent is the ratio of sine to cosine at a given angle — geometrically, the ratio of a right triangle's opposite side to its adjacent side. This page runs the FORWARD direction: given a known angle, it returns that angle's tangent value directly, useful whenever a formula in physics, engineering, or surveying calls for the tangent of an already-known angle.
This is the reverse of this site's Inverse Tangent (slope-to-angle) calculator, which instead starts from a rise-over-run ratio and works backward to find the angle that produced it — together, the two pages cover both directions of the identical relationship between an angle and its tangent.
Tangent grows without bound as the angle approaches 90°, since the adjacent side (tangent's denominator) shrinks toward 0 while the opposite side stays finite — a behavior sine and cosine, both bounded between −1 and 1, never show.
- Enter an angle into the Angle field.
- Read tan(angle): that angle's tangent value.
- Try 45° to see tangent land on exactly 1.
- Try an angle approaching 90° to see the tangent value grow rapidly without bound.
Worked example — 0°, 45°, and 60°
tan(45°)=1 exactly — the forward direction of this site's Inverse Tangent calculator, going from a known angle straight to its tangent value directly.
tan(0°)=0 — a flat angle has no rise at all relative to its run. tan(60°)=√3≈1.732, one of the exact values derivable from a 30-60-90 triangle's own fixed side ratios.
Questions
What is tangent?
The ratio of sine to cosine at a given angle — geometrically, a right triangle's opposite side divided by its adjacent side, the slope of the line that angle describes.
How is this different from the Inverse Tangent calculator on this site?
This page runs the FORWARD direction, from a known angle to its tangent value; that page runs the REVERSE, from a rise-over-run ratio back to the angle that produced it.
Why does tangent grow without bound near 90°?
Because tangent's denominator, the adjacent side (or equivalently, cosine), shrinks toward 0 as the angle approaches 90°, while the numerator stays finite — dividing by an ever-smaller number sends the result climbing without any upper limit.
What is tan(45°) and why is it exactly 1?
Exactly 1 — at 45°, sine and cosine are equal (both √2⁄2), and dividing any number by itself always gives 1.
Can tangent be negative?
Yes — beyond 90°, sine and cosine can carry opposite signs from each other, making their ratio, tangent, negative, unlike sine and cosine individually, each of which stays within a fixed −1-to-1 range.