IIT JEE MATHS

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

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

Voice and Writing issue solved on YouTube.

Solution 19 https://youtu.be/AYy8FKaV2O4

Q19 #mathematics @jee_mains

Solution 18 https://youtu.be/BvAm7N6JWsc

Question 18 #Mathematics @jee_mains

Solution 17 https://youtu.be/j284G1UrSc0

Question 17 #Mathematics @jee_mains

Solution 16 https://youtu.be/qYXuXWOJhWk

Solution 16 https://youtu.be/qYXuXWOJhWk