Trigonometric identities are tools for simplifying signal analysis and power calculations.

Unit circle with angles

Pythagorean Identity

$\sin^2\theta + \cos^2\theta = 1$

Double Angle Formulas

$\sin(2\theta) = 2\sin\theta\cos\theta$

$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1$

Sum/Difference Formulas

$\sin(\alpha \$

$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$

Product-to-Sum (signal mixing)

$\cos\alpha\cos\beta = \frac{1}{2}[\cos(\alpha-\beta) + \cos(\alpha+\beta)]$

EE Applications

Power calculations: For $v(t) = V_m\cos(\omega t)$ and $i(t) = I_m\cos(\omega t - \phi)$ :

$p(t) = \frac{V_m I_m}{2}[\cos\phi + \cos(2\omega t - \phi)]$

The DC term $\frac{V_m I_m}{2}\cos\phi$ is the average power.

RMS values: The identity $\cos^2\theta = \frac{1}{2}(1 + \cos 2\theta)$ explains why $V_{rms} = V_m/\sqrt{2}$ .
Phase shifts: $\cos\theta = \sin(\theta + 90°)$ — capacitor current leads voltage by 90°.

Fundamentals