\( \cos (a-b) \)

The expression \( \cos(a - b) \) can be expanded using the **cosine difference identity**. This identity is a fundamental trigonometric formula that expresses the cosine of the difference of two angles in terms of the sines and cosines of the individual angles. ## Cosine Difference Identity \[ \cos(a - b) = \cos a \cos b + \sin a \sin b \] ### Derivation The cosine difference identity can be derived from the **Euler's formula** or using the **unit circle** and trigonometric identities. Here's a simple derivation using the sum and difference formulas: 1. **Start with the sum identity:** \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \] 2. **Replace \( b \) with \( -b \):** \[ \cos(a - b) = \cos a \cos(-b) - \sin a \sin(-b) \] 3. **Use the even-odd properties of cosine and sine:** - \( \cos(-b) = \cos b \) (cosine is even) - \( \sin(-b) = -\sin b \) (sine is odd) 4. **Substitute these into the equation:** \[ \cos(a - b) = \cos a \cos b - \sin a (-\sin b) = \cos a \cos b + \sin a \sin b \] ### Example Suppose you want to find \( \cos(75^\circ) \). You can express 75° as the difference of two familiar angles, for example, \( 75^\circ = 45^\circ + 30^\circ \). Using the cosine difference identity: \[ \cos(75^\circ) = \cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ \] Plugging in the known values: \[ \cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \sin 30^\circ = \frac{1}{2} \] \[ \cos(75^\circ) = \left( \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} \right) - \left( \frac{\sqrt{2}}{2} \times \frac{1}{2} \right) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \] ### Applications Understanding the cosine difference identity is essential in various fields such as: - **Solving trigonometric equations:** It allows the simplification of complex trigonometric expressions. - **Signal processing:** Used in the analysis of waveforms and oscillations. - **Engineering and Physics:** Helps in solving problems involving harmonic motion and oscillatory systems. ### Summary The cosine of the difference of two angles can be expressed as: \[ \cos(a - b) = \cos a \cos b + \sin a \sin b \] This identity is a powerful tool in simplifying and solving a wide range of trigonometric problems.