Class 12th maths notes

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❌ 𝐑𝐞𝐯𝐞𝐚𝐥𝐢𝐧𝐠 𝗠𝘆 "𝗣𝗔𝗜𝗗" 𝗰𝗵𝗮𝗻𝗻𝗲𝗹 𝗮𝘀 𝗲𝘅𝗮𝗺𝘀 𝗮𝗿𝗲 𝗻𝗲𝗮𝗿 ❌ 👉🏻𝐨𝐧𝐥𝐲 𝐟𝐨𝐫 7 𝐦𝐢𝐧𝐬 ⏰ 👉🏻𝐥𝐢𝐧𝐤 𝐰𝐢𝐥𝐥 𝐞𝐱𝐩𝐢𝐫𝐞 & 𝐨𝐧𝐥𝐲 1𝐬𝐭 32 𝐚𝐬𝐩𝐢𝐫𝐚𝐧𝐭𝐬 𝐚𝐥𝐥𝐨𝐰𝐞𝐝⚠️ 🔗🔗1st 𝗟𝗶𝗻𝗸 👉🏻👉🏻 https://t.me/+5BryLO8Aq0U2ZDll https://t.me/+5BryLO8Aq0U2ZDll https://t.me/+5BryLO8Aq0U2ZDll 2nd link JEE IHL https://t.me/+AWOR3nmyBN4xYTU9 https://t.me/+AWOR3nmyBN4xYTU9 https://t.me/+AWOR3nmyBN4xYTU9 3rd link NEET IHL https://t.me/+Ad9gmx4pBKw4NGJl https://t.me/+Ad9gmx4pBKw4NGJl https://t.me/+Ad9gmx4pBKw4NGJl 4th link https://t.me/+fDTyAwmk54VhZmNl https://t.me/+fDTyAwmk54VhZmNl https://t.me/+fDTyAwmk54VhZmNl #𝙨𝙚𝙘𝙧𝙚𝙩_𝙘𝙝𝙖𝙣𝙣𝙚𝙡🗳🗳😎😱 Note - last tym many could not join our secret channel😅 so last tym publishing the link‼️ ⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️ Not a promotion, channel for your benefit Join fast else regret later😌😉

❌ 𝐑𝐞𝐯𝐞𝐚𝐥𝐢𝐧𝐠 𝗠𝘆 "𝗣𝗔𝗜𝗗" 𝗰𝗵𝗮𝗻𝗻𝗲𝗹 𝗮𝘀 𝗲𝘅𝗮𝗺𝘀 𝗮𝗿𝗲 𝗻𝗲𝗮𝗿 ❌ 👉🏻𝐨𝐧𝐥𝐲 𝐟𝐨𝐫 7 𝐦𝐢𝐧𝐬 ⏰ 👉🏻𝐥𝐢𝐧𝐤 𝐰𝐢𝐥𝐥 𝐞𝐱𝐩𝐢𝐫𝐞 & 𝐨𝐧𝐥𝐲 1𝐬𝐭 32 𝐚𝐬𝐩𝐢𝐫𝐚𝐧𝐭𝐬 𝐚𝐥𝐥𝐨𝐰𝐞𝐝⚠️ 🔗🔗1st 𝗟𝗶𝗻𝗸 👉🏻👉🏻 https://t.me/+5BryLO8Aq0U2ZDll https://t.me/+5BryLO8Aq0U2ZDll https://t.me/+5BryLO8Aq0U2ZDll 2nd link JEE IHL https://t.me/+AWOR3nmyBN4xYTU9 https://t.me/+AWOR3nmyBN4xYTU9 https://t.me/+AWOR3nmyBN4xYTU9 3rd link NEET IHL https://t.me/+Ad9gmx4pBKw4NGJl https://t.me/+Ad9gmx4pBKw4NGJl https://t.me/+Ad9gmx4pBKw4NGJl 4th link https://t.me/+fDTyAwmk54VhZmNl https://t.me/+fDTyAwmk54VhZmNl https://t.me/+fDTyAwmk54VhZmNl #𝙨𝙚𝙘𝙧𝙚𝙩_𝙘𝙝𝙖𝙣𝙣𝙚𝙡🗳🗳😎😱 Note - last tym many could not join our secret channel😅 so last tym publishing the link‼️ ⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️ Not a promotion, channel for your benefit Join fast else regret later😌😉

❌ 𝐑𝐞𝐯𝐞𝐚𝐥𝐢𝐧𝐠 𝗠𝘆 "𝗣𝗔𝗜𝗗" 𝗰𝗵𝗮𝗻𝗻𝗲𝗹 𝗮𝘀 𝗲𝘅𝗮𝗺𝘀 𝗮𝗿𝗲 𝗻𝗲𝗮𝗿 ❌ 👉🏻𝐨𝐧𝐥𝐲 𝐟𝐨𝐫 7 𝐦𝐢𝐧𝐬 ⏰ 👉🏻𝐥𝐢𝐧𝐤 𝐰𝐢𝐥𝐥 𝐞𝐱𝐩𝐢𝐫𝐞 & 𝐨𝐧𝐥𝐲 1𝐬𝐭 32 𝐚𝐬𝐩𝐢𝐫𝐚𝐧𝐭𝐬 𝐚𝐥𝐥𝐨𝐰𝐞𝐝⚠️ 🔗🔗1st 𝗟𝗶𝗻𝗸 👉🏻👉🏻 https://t.me/+JkhOyr9yTBU1ZjZl https://t.me/+JkhOyr9yTBU1ZjZl https://t.me/+JkhOyr9yTBU1ZjZl 2nd link JEE IHL https://t.me/+AWOR3nmyBN4xYTU9 https://t.me/+AWOR3nmyBN4xYTU9 https://t.me/+AWOR3nmyBN4xYTU9 3rd link NEET IHL https://t.me/+Ad9gmx4pBKw4NGJl https://t.me/+Ad9gmx4pBKw4NGJl https://t.me/+Ad9gmx4pBKw4NGJl 4th link https://t.me/+fDTyAwmk54VhZmNl https://t.me/+fDTyAwmk54VhZmNl https://t.me/+fDTyAwmk54VhZmNl #𝙨𝙚𝙘𝙧𝙚𝙩_𝙘𝙝𝙖𝙣𝙣𝙚𝙡🗳🗳😎😱 Note - last tym many could not join our secret channel😅 so last tym publishing the link‼️ ⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️ Not a promotion, channel for your benefit Join fast else regret later😌😉

❌ 𝐑𝐞𝐯𝐞𝐚𝐥𝐢𝐧𝐠 𝗠𝘆 "𝗣𝗔𝗜𝗗" 𝗰𝗵𝗮𝗻𝗻𝗲𝗹 𝗮𝘀 𝗲𝘅𝗮𝗺𝘀 𝗮𝗿𝗲 𝗻𝗲𝗮𝗿 ❌ 👉🏻𝐨𝐧𝐥𝐲 𝐟𝐨𝐫 7 𝐦𝐢𝐧𝐬 ⏰ 👉🏻𝐥𝐢𝐧𝐤 𝐰𝐢𝐥𝐥 𝐞𝐱𝐩𝐢𝐫𝐞 & 𝐨𝐧𝐥𝐲 1𝐬𝐭 32 𝐚𝐬𝐩𝐢𝐫𝐚𝐧𝐭𝐬 𝐚𝐥𝐥𝐨𝐰𝐞𝐝⚠️ 🔗🔗1st 𝗟𝗶𝗻𝗸 👉🏻👉🏻 https://t.me/+_gBAwuoMJtpmMjY1 https://t.me/+_gBAwuoMJtpmMjY1 https://t.me/+_gBAwuoMJtpmMjY1 2nd link JEE IHL https://t.me/+QuQJHHGiBb8wOTll https://t.me/+QuQJHHGiBb8wOTll https://t.me/+QuQJHHGiBb8wOTll 3rd link NEET IHL https://t.me/+_gBAwuoMJtpmMjY1 https://t.me/+_gBAwuoMJtpmMjY1 https://t.me/+_gBAwuoMJtpmMjY1 4th link https://t.me/+QuQJHHGiBb8wOTll https://t.me/+QuQJHHGiBb8wOTll https://t.me/+QuQJHHGiBb8wOTll #𝙨𝙚𝙘𝙧𝙚𝙩_𝙘𝙝𝙖𝙣𝙣𝙚𝙡🗳🗳😎😱 Note - last tym many could not join our secret channel😅 so last tym publishing the link‼️ ⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️ Not a promotion, channel for your benefit Join fast else regret later😌😉

📚Notes on Trigonometric Equations and Identities📚 A function f(x) is said to be periodic if there exists some T > 0 such that f(x+T) = f(x) for all x in the domain of f(x). In case, the T in the definition of period of f(x) is the smallest positive real number then this ‘T’ is called the period of f(x). Periods of various trigonometric functions are listed below: 1) sin x has period 2π 2) cos x has period 2π 3) tan x has period π 4) sin(ax+b), cos (ax+b), sec(ax+b), cosec (ax+b) all are of period 2π/a 5) tan (ax+b) and cot (ax+b) have π/a as their period 6) |sin (ax+b)|, |cos (ax+b)|, |sec(ax+b)|, |cosec (ax+b)| all are of period π/a 7) |tan (ax+b)| and |cot (ax+b)| have π/2a as their period ➖Sum and Difference Formulae of Trigonometric Ratios 1) sin(a + ß) = sin(a)cos(ß) + cos(a)sin(ß) 2) sin(a – ß) = sin(a)cos(ß) – cos(a)sin(ß) 3) cos(a + ß) = cos(a)cos(ß) – sin(a)sin(ß) 4) cos(a – ß) = cos(a)cos(ß) + sin(a)sin(ß) 5) tan(a + ß) = [tan(a) + tan (ß)]/ [1 - tan(a)tan (ß)] 6)tan(a - ß) = [tan(a) - tan (ß)]/ [1 + tan (a) tan (ß)] 7) tan (π/4 + θ) = (1 + tan θ)/(1 - tan θ) 8) tan (π/4 - θ) = (1 - tan θ)/(1 + tan θ) 9) cot (a + ß) = [cot(a) . cot (ß) - 1]/ [cot (a) +cot (ß)] 10) cot (a - ß) = [cot(a) . cot (ß) + 1]/ [cot (ß) - cot (a)] ➖Double or Triple -Angle Identities 1) sin 2x = 2sin x cos x 2) cos2x = cos2x – sin2x = 1 – 2sin2x = 2cos2x – 1 3) tan 2x = 2 tan x / (1-tan 2x) 4) sin 3x = 3 sin x – 4 sin3x 5) cos3x = 4 cos3x – 3 cosx 6) tan 3x = (3 tan x - tan3x) / (1- 3tan 2x) ➖For angles A, B and C, we have 1) sin (A + B +C) = sinAcosBcosC + cosAsinBcosC + cosAcosBsinC - sinAsinBsinC 2) cos (A + B +C) = cosAcosBcosC- cosAsinBsinC - sinAcosBsinC - sinAsinBcosC 3) tan (A + B +C) = [tan A + tan B + tan C –tan A tan B tan C]/ [1- tan Atan B - tan B tan C –tan A tan C 4) cot (A + B +C) = [cot A cot B cot C – cotA - cot B - cot C]/ [cot A cot B + cot Bcot C + cot A cotC–1] ➖List of some other trigonometric formulas: 1) 2sinAcosB = sin(A + B) + sin (A - B) 2) 2cosAsinB = sin(A + B) - sin (A - B) 3) 2cosAcosB = cos(A + B) + cos(A - B) 4) 2sinAsinB = cos(A - B) - cos (A + B) 5) sin A + sin B = 2 sin [(A+B)/2] cos [(A-B)/2] 6) sin A - sin B = 2 sin [(A-B)/2] cos [(A+B)/2] 7) cosA + cos B = 2 cos [(A+B)/2] cos [(A-B)/2] 8) cosA - cos B = 2 sin [(A+B)/2] sin [(B-A)/2] 9) tanA ± tanB = sin (A ± B)/ cos A cos B 10)cot A ± cot B = sin (B ± A)/ sin A sin B ➖Method of solving a trigonometric equation: 1) If possible, reduce the equation in terms of any one variable, preferably x. Then solve the equation as you used to in case of a single variable. 2) Try to derive the linear/algebraic simultaneous equations from the given trigonometric equations and solve them as algebraic simultaneous equations. 3) At times, you might be required to make certain substitutions. It would be beneficial when the system has only two trigonometric functions. ➖Some results which are useful for solving trigonometric equations: 1) sin θ = sina and cosθ = cosa ⇒ θ = 2nπ + a 2) sin θ = 0 ⇒ θ = nπ 3) cosθ = 0 ⇒ θ = (2n + 1)π/2 4) tan θ = 0 ⇒ θ = nπ 5) sinθ = sina⇒ θ = nπ + (-1)na where a ∈ [–π/2, π/2] 6) cosθ= cos a ⇒ θ = 2nπ ± a, where a ∈[0,π] 7) tanθ = tana⇒ θ = nπ+ a, where a ∈[–π/2, π/2] 8) sinθ = 1 ⇒ θ= (4n + 1)π/2 9) sin θ = -1 ⇒ θ = (4n - 1) π /2 10) sin θ = -1 ⇒ θ = (2n +1) π /2 11) |sinθ| = 1⇒ θ =2nπ 12) cosθ = 1 ⇒ θ =(2n + 1) 13) |cosθ| = 1⇒ θ =nπ Credit - @DCPANDEYBOOKFREE @IRODOV_dc_pandey

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