Reduction Formulas in Trigonometry
Big Idea: Reduction formulas allow us to express trigonometric functions of any angle in terms of a trigonometric function of an acute angle (an angle between $0°$ and $90°$). This is incredibly powerful because it means we only need to know trig values for angles in the first quadrant!
🔑 Why Do We Need Reduction Formulas?
Imagine you need to find $\sin(150°)$. Your calculator can do it, but understanding why $\sin(150°) = \sin(30°) = \frac{1}{2}$ requires reduction formulas.
These formulas "reduce" angles from any quadrant back to the first quadrant (acute angles), while adjusting the sign based on the CAST rule.
📐 The CAST Diagram
The CAST rule tells us which trig functions are positive in each quadrant:
- Quadrant I (0°–90°): All positive
- Quadrant II (90°–180°): Sin positive
- Quadrant III (180°–270°): Tan positive
- Quadrant IV (270°–360°): Cos positive