Lesson 3.11

Quadrantal Angles

Quadrantal angles lie exactly on the axes — they don't belong to any quadrant at all. Their trig values are the simplest to memorize, but they have some surprising results involving zero and “undefined.”

Introduction

The four quadrantal angles — (or ) — are the compass points of the unit circle. Their trig values come directly from reading the coordinates of the points where the circle crosses the axes.

Past Knowledge

On the unit circle, every point is .

Today's Goal

Know all six trig values (or identify them as undefined) for the four quadrantal angles.

Future Success

These are the “anchor points” of every trig graph. They define the intercepts, maxima, minima, and asymptotes.

Key Concepts

The Four Quadrantal Angles

A quadrantal angle is one whose terminal side lies along an axis. The unit circle coordinates give us sine and cosine directly:

Angle	Radians	Point (x, y)	cos θ	sin θ	tan θ

					undef

					undef

Why is Tangent Sometimes Undefined?

Since , tangent is undefined whenever the denominator . This happens at and (where the terminal side sits on the -axis).

Pattern to Memorize

Cosine reads the -axis:
Sine reads the -axis:

Worked Examples

Basic

Direct Lookup

Question: Find and .

Step 1: The angle radians () points to on the unit circle.

Step 2: Read off the coordinates: , .

Final Answer: ,

Intermediate

Undefined Detection

Question: Find .

Step 1: At , the point is .

Step 2: — division by zero!

Final Answer: is undefined.

Advanced

Coterminal Quadrantal

Question: Find .

Step 1: Find the coterminal angle.

Step 2: At , the point is .

Final Answer:

Common Pitfalls

Writing 0 Instead of Undefined

is undefined, not . When the denominator is zero, the expression has no value. Writing is actively wrong and will cost points.

❌

✅ is undefined

Real-Life Applications

Cardinal Directions in Navigation

The four quadrantal angles correspond to the four cardinal directions on a compass: East (), North (), West (), South (). When autopilot systems compute heading adjustments, they use the cosine component (east-west displacement) and sine component (north-south displacement) directly — and at the cardinals, one component is always zero.