Tangent

Tangent measures the ratio of vertical change to horizontal change: opposite divided by adjacent.

Relative to θ, sine, cosine, and tangent compare opposite, adjacent, and hypotenuse sides. A right triangle has one acute angle labelled theta. The vertical side is opposite theta, the horizontal leg is adjacent, and the sloping longest side is the hypotenuse; a square marks the ninety-degree angle.

Tangent compares rise with run, connecting right triangles, slope, angles, and real-world steepness.

Meaning in a right triangle

For an acute angle $\theta$ in a right triangle,

\tan\theta=\frac{\text{opposite}}{\text{adjacent}}.

This is TOA in SOH–CAH–TOA. Tangent is the only primary right-triangle ratio that does not use the hypotenuse. It is often the most direct choice when a problem gives a vertical height and horizontal distance.

Worked example: find a height

Worked example: find a horizontal distance

A wheelchair ramp rises $0.75$ m and makes an angle of $5^\circ$ with the ground. Let $d$ be the horizontal run:

\tan5^\circ=\frac{0.75}{d}.

Rearrange:

d=\frac{0.75}{\tan5^\circ}\approx8.57\text{ m}.

A shallow angle creates a long run, so the magnitude makes sense.

Finding an angle with inverse tangent

If opposite and adjacent are known, use inverse tangent:

\tan\theta=\frac{7}{9},

\theta=\tan^{-1}\left(\frac79\right)\approx37.9^\circ.

In coordinate geometry, a line with slope $7/9$ therefore makes an inclination of about $37.9^\circ$ above the positive horizontal axis.

Common mistakes

Using opposite over hypotenuse. That is sine; tangent uses opposite over adjacent.

Measuring the angle from vertical. Elevation and inclination are normally measured from horizontal.

Treating $\tan^{-1}$ as a reciprocal. Inverse tangent finds an angle; $1/\tan$ is cotangent.

Equating a $12\%$ grade with $12^\circ$. Convert the percentage to a ratio, then use inverse tangent.

Forgetting observer height. Decide whether the right triangle starts at ground level or eye level.

Quick self-check

Did I choose and mark the reference angle?
Are opposite and adjacent labelled relative to it?
Does the quotient represent rise divided by run?
Am I using tangent for a side or inverse tangent for an angle?
Is the calculator in degree mode?
Does the steepness match the size of the answer?