Introduction
The unit circle is infinite, but we only need to memorize one quadrant. The rest is just symmetry and signs.
Past Knowledge
This relies on the 30-60-90 and 45-45-90 special right triangles you learned in Geometry.
Today's Goal
We use the "Bowtie Method" (drawing reference triangles to the x-axis) to find values anywhere on the circle, adjusting signs based on the quadrant.
Future Success
You will need to evaluate these instantly in Calculus. requires knowing without a calculator.
Key Concepts
The First Quadrant (Memorize This)
| Degrees | Radians | Cos (x) | Sin (y) |
|---|---|---|---|
All Students Take Calculus (ASTC)
A mnemonic for knowing which functions are positive in which quadrant:
Worked Examples
Example 1: Quadrant II
Evaluate .
Find Reference Angle
To get to , we need more. So calculation is based on .
Check Sign
is in Quadrant II. Sine (y-coord) is positive there.
.
Example 2: Quadrant IV (Radians)
Evaluate .
Find Reference Angle
Denominator is 3. The "family" is (which is ).
Check Sign
is just short of . So it's in Quadrant IV.
Cosine is Positive in Q4.
Example 3: Negative Angles
Evaluate .
Locate Quadrant
Rotate clockwise. lands in Quadrant III.
Determine Sign and Value
Tangent is Positive in Q3.
Ref angle is ().
.
Common Pitfalls
Confusing 30 and 60
This is the most common mistake. but . Remember: 30 degrees is "shallow", so it has a BIG x-value ().
Real-World Application
Hexagonal Tessellation
Bees build honeycombs in hexagons because they are efficient. The geometry of a hexagon is built entirely on triangles. Structural engineers use these exact ratios to calculate load distribution in truss bridges.
Practice Quiz
Practice Quiz
Loading...