The Cosine Combination
The sum and difference formulas for Cosine are slightly different from Sine. Notice the sign flip in the middle!
\[\cos(A + B) = \cos A \cos B - \sin A \sin B\]
\[\cos(A - B) = \cos A \cos B + \sin A \sin B\]
Worked Examples
Example 1: Evaluating Cos(15°)
Find the exact value of \(\cos(15^\circ)\).
- \(15^\circ = 45^\circ - 30^\circ\).
- \(\cos(45-30) = \cos(45)\cos(30) + \sin(45)\sin(30)\).
- \(= (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2})\).
- \(= \frac{\sqrt{6} + \sqrt{2}}{4} \approx 0.966\).
Example 2: Identity Proof
Show that \(\cos(x - \pi/2) = \sin(x)\).
- \(\cos(x - \pi/2) = \cos(x)\cos(\pi/2) + \sin(x)\sin(\pi/2)\).
- Since \(\cos(\pi/2) = 0\) and \(\sin(\pi/2) = 1\):
- \(= \cos(x)(0) + \sin(x)(1) = \sin(x)\).
- Result: Proof complete.
The Bridge to Quantum Mechanics
In Quantum Mechanics, the Probability Amplitude of a particle moving from point A to point B is often represented by a "Cosine" wave. When we have multiple paths, the total amplitude is the sum of these waves. The "sign flip" in the cosine sum formula is the mathematical reason why certain paths Cancel each other out (destructive interference). This is how we prove that light travels in a straight line; the "bent" paths have phases that cancel out due to these exact trig identities.