Lesson 3.12

The First Quadrant

The first quadrant is the master key to the entire unit circle. Memorize the five angles here, and you can derive every other angle using symmetry.

Introduction

Everything in the unit circle begins with Quadrant I. This is the region where both and are positive, so both cosine and sine are positive. It contains five key angles between and , and their coordinates come entirely from the two special triangles you just learned.

Past Knowledge

45-45-90 gives . 30-60-90 gives and .

Today's Goal

Memorize all Quadrant I unit circle coordinates and their corresponding angles.

Future Success

With reference angles (next lesson), you'll mirror these five values into the other three quadrants to fill the entire circle.

Key Concepts

The Five First Quadrant Angles

Degrees	Radians	(cos θ, sin θ)	Source
			Quadrantal
			30-60-90
			45-45-90
			30-60-90
			Quadrantal

The Memorization Pattern

Notice the cosine values decrease as the angle increases: .

Meanwhile, the sine values increase: .

The “Counting” Trick

Write the numerators as a sequence: — all over .

Since , , and , the sequence simplifies to: .

This is the sine sequence for 0°, 30°, 45°, 60°, 90°. Cosine is the same sequence in reverse.

Worked Examples

Basic

Quick Recall

Question: Find and .

Step 1: . From the table: .

Final Answer: ,

Intermediate

Computing Tangent from the Table

Question: Find .

Step 1: At , the point is .

Step 2:

Final Answer:

Advanced

Evaluating an Expression

Question: Evaluate .

Step 1: Substitute values.

Step 2: Compute.