Tangent Ratio Calculator

Learn how to calculate the tangent ratio in a right triangle with this handy tool. Discover the relationship between the tangent trigonometric function and triangles, how to find (when possible) the tangent ratio, and much more!

The tangent ratio is a way to define the trigonometric function tangent in right triangles* — and not only. We define the tangent ratio as a way to calculate the tangent of an angle when the angle itself is unknown.

To define the tangent ratio, we need to consider a triangle, a right triangle to be precise. In it, we identify:

• Two acute angles (say, $\alpha$ and $\beta$);
• Two sides that form a right angle at their intersection, the legs of the triangle; and
• A right angle and the hypotenuse opposite to it.

Each angle is associated with an opposite and an adjacent leg, and their roles switch when we consider the other angle. The opposite leg is the one with which the angle doesn't share a side, while the adjacent one corresponds to one of the angles' sides. Now that you identified the elements of a right triangle, we can answer the question: "what is the tangent ratio?.

The tangent ratio is the ratio between the opposite and the adjacent leg. This quantity also corresponds to the tangent of the angle we used in identifying the legs. Writing this as a formula, we have:

From this simple formula, we can already point out some features of the tangent function:

• When the adjacent leg goes to $0$, the tangent is undefined. This condition corresponds to what we find when the angle $\alpha$ is equal to $90\degree$.
• Since the tangent ratio is defined within right triangles, we also notice that it can only assume positive values: this tells us that we only need to know the results for angles between $0\degree$ and $90\degree$ to figure out all the others around the circle.

To calculate the tangent ratio, you must know the length of the legs of the right triangle you're working on. If you don't know the length of a leg, use the Pythagorean theorem to obtain it from the hypotenuse.

Once you know the length of the legs, calculate both the ratios: your results are the tangents of the two acute angles of the right triangle. If you want to calculate the angles, use the inverse tangent function!

If the tangent ratio calculator stimulated your curiosity and you want to learn more about trigonometric function, don't miss our other tools:

• The trigonometry calculator;
• The cosine triangle calculator;
• The sine triangle calculator;
• The trig triangle calculator;
• The right triangle trigonometry calculator;
• The trig calculator;
• The sine cosine tangent calculator; and
• The tangent angle calculator.