Trigonometry

Trigonometry is based on the Right-Angled Triangle. Remember PBH (Perpendicular combined with Base and Hypotenuse).

• $\sin \theta = P/H$
• $\cos \theta = B/H$
• $\tan \theta = P/B$

The Master Table (0 to 90): Memorize values of Sin. Cos is reverse of Sin. Tan = Sin/Cos.

This rule decides if a value is Positive or Negative depending on the Quadrant.

Transformation Rule:

• 90/270 (Odd Multiples): Change Function (Sin $\leftrightarrow$ Cos, Tan $\leftrightarrow$ Cot).
• 180/360 (Even Multiples): No Change.
• Example: $\sin(90+\theta) = \cos \theta$ (2nd Quad, Sin is +ve).

These are the tools to simplify complex expressions.

• $\sin^2\theta + \cos^2\theta = 1$
• $1 + \tan^2\theta = \sec^2\theta$ (Note: $\sec^2 - \tan^2 = 1 \Rightarrow (\sec-\tan)(\sec+\tan)=1$)
• $1 + \cot^2\theta = \text{cosec}^2\theta$

Pythagorean Triplets (Shortcut for H/D):

Convert hard questions into 5-second arithmetic.

• Form involves sin/cos only: Put $\theta = 0^\circ$ or $90^\circ$.
• Form involves tan/cot/sec/cosec: Put $\theta = 45^\circ$ (Avoid undefined terms like $\tan 90$).
• If options are numerical: Puttings works 100%.
• Condition: Denominator must basic Identity not become zero.

Use Ratios instead of Tan formulas.

• 30-60-90 Triangle: Side opposite to $30^\circ$ is $1$, $60^\circ$ is $\sqrt{3}$, $90^\circ$ is $2$.
• 45-45-90 Triangle: Side opposite to $45^\circ$ is $1$, Hypotenuse is $\sqrt{2}$.
• Broken Tree: Total height = Base + Hypotenuse.
• Shadow Ratio: Height : Shadow = $1 : \sqrt{3}$ means Angle is $30^\circ$.