Sum and Difference Formulas for Trigonometric Functions
Big Idea: The sum and difference formulas allow us to find the exact values of trigonometric functions for angles that can be expressed as the sum or difference of known angles (like 30°, 45°, 60°, 90°). They are among the most powerful tools in trigonometry!
📐 Why Do We Need These Formulas?
You already know the exact values of $\sin$, $\cos$, and $\tan$ for special angles like $30°$, $45°$, $60°$, $90°$, etc. But what about angles like $75°$ or $15°$?
Notice that $75° = 45° + 30°$ and $15° = 45° - 30°$. If we had a formula to combine these, we could find exact values for many more angles!
⚠️ Common Mistake: Many students assume $\sin(A + B) = \sin A + \sin B$. This is FALSE! For example, $\sin(30° + 60°) = \sin 90° = 1$, but $\sin 30° + \sin 60° = \frac{1}{2} + \frac{\sqrt{3}}{2} \approx 1.366$. They are NOT equal!