Tangent

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,

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.

Tangent as rise over run

If a line makes angle $\theta$ with the positive horizontal direction, a right triangle drawn along the line has

That quotient is also the line's slope, so

A steeper positive line has a larger angle and a larger tangent value. This connection lets geometry, trigonometry, and coordinate algebra describe the same rate of change.

When to use tangent

Use tangent when the relevant sides are opposite and adjacent. Choose the reference angle first, label the sides relative to it, and ignore the hypotenuse if it is not known or required.

If the useful pair includes the hypotenuse, choose sine or cosine instead.

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:

Rearrange:

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:

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

How tangent changes

For acute angles, tangent is positive. It starts near $0$ for an angle near $0^\circ$ and grows rapidly as the angle approaches $90^\circ$.

Key exact values are

At $90^\circ$, the adjacent horizontal run is $0$, so tangent would require division by zero. Therefore $\tan90^\circ$ is undefined.

Angle of elevation and depression

An angle of elevation is measured upward from a horizontal line. An angle of depression is measured downward from a horizontal line. Since separate horizontal lines are parallel, alternate interior angles often let an angle of depression at the observer equal an angle of elevation at the object.

Draw and label the horizontal line explicitly. Measuring from a wall or vertical line instead changes the reference angle and the side labels.

Percent grade and angle

Road and ramp grade is commonly expressed as

Because rise/run is tangent, a $10\%$ grade has $\tan\theta=0.10$, so $\theta=\tan^{-1}(0.10)\approx5.7^\circ$. A percent grade and an angle in degrees are related, but they are not the same number.

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?

Related topics

• Right Triangle Trigonometry
• Sine
• Cosine
• Slope
• Angles of Elevation and Depression