Detailed definition
Understanding Tangent
In right-triangle trigonometry, tangent of an acute angle is the ratio of the opposite side to the adjacent side. Unlike sine and cosine, tangent does not use the hypotenuse.
Because tangent compares the two legs, it is especially useful when the right triangle is read as a rise-over-run style shape. That makes tangent feel closely related to slope in many geometric settings.
This page keeps the angle, opposite leg, and adjacent leg visible together so tangent is read from a specific angle-based structure rather than from memorised letters alone.
Key facts
Important ideas to remember
- Tangent is the ratio of opposite side to adjacent side in a right triangle.
- Tangent is commonly remembered with TOA: opposite over adjacent.
- Tangent can also be written as sine divided by cosine for the same angle.
- In many right-triangle settings, tangent describes how steeply the triangle rises relative to horizontal run.
Where it is used
Where tangent shows up
- Use tangent when the known or unknown sides are the opposite and adjacent legs of a right triangle.
- Use it in angle-of-elevation and angle-of-depression problems where rise and run matter.
- Use tangent when comparing right-triangle ratios to slope-style reasoning.
Common mistakes
What to watch out for
- Do not include the hypotenuse in the tangent ratio.
- Do not assign opposite and adjacent without first fixing the reference angle.
- Do not confuse tangent as a trig ratio here with tangent as a line touching a circle in a different geometry context.