Introduction
We defined and . But what are the actual numbers? For most angles, they are ugly decimals. But for "Special" angles (multiples of 30, 45, 60), exact values exist.
Prerequisite Connection
This relies on the 30-60-90 and 45-45-90 special right triangles you learned in Geometry.
Today's Increment
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.
Why This Matters
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
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