Lesson 8.2.4

The Tangent Ratio

The tangent ratio () uses only the two legsof a right triangle — no hypotenuse needed. It's ideal when you know one leg and an angle and want the other leg.

Introduction

Tangent is the ratio of the opposite leg to the adjacent leg. It's the “rise over run” of the hypotenuse — which is exactly why slope is the tangent of the angle of inclination.

Past Knowledge

SOH CAH TOA (8.2.3). Special triangles (8.2.1–8.2.2). Slope.

Today's Goal

Solve for missing legs using the tangent ratio.

Future Success

Sine & cosine (8.2.5), angles of elevation/depression (8.3.2).

Key Concepts

Tangent Formula

Use tangent when you have (or want) the two legs and don't care about the hypotenuse.

Solving for a Missing Side

Find opposite:
Find adjacent:

Worked Examples

Basic

Finding the Opposite

Angle = 35°, adjacent = 10. Find the opposite side.

Intermediate

Finding the Adjacent

Angle = 50°, opposite = 15. Find the adjacent side.

Advanced

Slope as Tangent

A ramp rises 6 ft over a horizontal distance of 40 ft. What angle does it make with the ground?

Common Pitfalls

Calculator in Wrong Mode

Make sure your calculator is in degree mode, not radians. should give 1. If you get 1.6198, you're in radian mode.

Using Tan When You Need Sin or Cos

If the problem involves the hypotenuse, use sine or cosine — tangent only relates the two legs.

Real-Life Applications

ADA Ramp Requirements

The ADA requires ramps with a maximum slope of 1:12 (1 inch rise per 12 inches run). That's , giving a maximum angle of about 4.76°.

Surveying — Grade Percentage

Road signs that say “6% grade” mean , or a rise of 6 feet per 100 feet of horizontal distance. That's a 3.4° incline.