The Master Formula Box
Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$Compound Angle: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$Double Angle: $\cos 2\theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$
Dimensional Analysis of Trig Functions
Trigonometric functions take a dimensionless ratio (an angle) as an input and produce a dimensionless ratio (side/side) as an output. Crucially: The argument inside a trig function in a Physics formula (like $\omega t$) must always be dimensionless. If $t$ is in seconds, $\omega$ must be in $s^{-1}$ (rad/s).
Variations: Inverse Trigonometry
In JEE, we often need to find the angle from a value: $\theta = \sin^{-1}(x)$.Key Range: $\sin^{-1}(x)$ and $\tan^{-1}(x)$ are defined in $[-\pi/2, \pi/2]$, while $\cos^{-1}(x)$ is defined in $[0, \pi]$. Misunderstanding these ranges is a primary cause of marks loss in Inverse Trig questions.
Shortcuts & Mnemonics
- SOH CAH TOA: Sine = Opposite/Hypotenuse; Cosine = Adjacent/Hypotenuse; Tangent = Opposite/Adjacent.
- The $\tan 15^\circ$ Trick: $\tan 15^\circ = 2 - \sqrt{3}$ and $\tan 75^\circ = 2 + \sqrt{3}$. These appear constantly in JEE; memorizing them saves 2 minutes.
- All Silver Tea Cups: To remember the positive ratios in quadrants 1, 2, 3, and 4.
Edge Cases
- $\theta = 90^\circ$ for Tangent: $\tan 90^\circ$ is undefined ($\infty$). Physically, this represents a vertical slope.
- Small Angle Approximation: For $\theta < 10^\circ$, $\sin \theta \approx \theta$ and $\tan \theta \approx \theta$ (where $\theta$ is in radians). This is used extensively in Ray Optics and Pendulum problems.
- $\sin \theta = 0$: General solution is $\theta = n\pi$.
- $\cos \theta = 0$: General solution is $\theta = (2n+1)\pi/2$.
