12TH CLASS MATHS

📌 Basic Trigonometric Ratios

sin⁡θ=OppositeHypotenuse,cos⁡θ=AdjacentHypotenuse,tan⁡θ=sin⁡θcos⁡θ\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}, \quad \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta}sinθ=HypotenuseOpposite,cosθ=HypotenuseAdjacent,tanθ=cosθsinθ csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ\csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta}cscθ=sinθ1,secθ=cosθ1,cotθ=tanθ1

📌 Pythagorean Identities

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1sin2θ+cos2θ=1 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta1+tan2θ=sec2θ 1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta1+cot2θ=csc2θ

📌 Angle Transformation

sin⁡(−θ)=−sin⁡θ,cos⁡(−θ)=cos⁡θ\sin(-\theta) = -\sin \theta, \quad \cos(-\theta) = \cos \thetasin(−θ)=−sinθ,cos(−θ)=cosθ tan⁡(−θ)=−tan⁡θ\tan(-\theta) = -\tan \thetatan(−θ)=−tanθ

📌 Standard Angles

📌 Sum and Difference Formulas

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A \cos B \pm \cos A \sin Bsin(A±B)=sinAcosB±cosAsinB cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A \cos B \mp \sin A \sin Bcos(A±B)=cosAcosB∓sinAsinB tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}tan(A±B)=1∓tanAtanBtanA±tanB

📌 Double Angle Formulas

sin⁡2A=2sin⁡Acos⁡A\sin 2A = 2 \sin A \cos Asin2A=2sinAcosA cos⁡2A=cos⁡2A−sin⁡2A=2cos⁡2A−1=1−2sin⁡2A\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 Acos2A=cos2A−sin2A=2cos2A−1=1−2sin2A tan⁡2A=2tan⁡A1−tan⁡2A\tan 2A = \frac{2\tan A}{1 - \tan^2 A}tan2A=1−tan2A2tanA

📌 Product to Sum Formulas

sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)]\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]sinAsinB=21[cos(A−B)−cos(A+B)] cos⁡Acos⁡B=12[cos⁡(A−B)+cos⁡(A+B)]\cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]cosAcosB=21[cos(A−B)+cos(A+B)] sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)]sinAcosB=21[sin(A+B)+sin(A−B)]