Thursday 16 May 2019

Important Short Tricks on Trigonometric indenties



Important Short Tricks on Trigonometric.  Indenties


Pythagorean Identities


sin2 θ + cos2 θ = 1


tan2 θ + 1 = sec2 θ


cot2 θ + 1 = csc2 θ


Negative of a Function


sin (–x) = –sin x


cos (–x) = cos x


tan (–x) = –tan x


csc (–x) = –csc x


sec (–x) = sec x


cot (–x) = –cot x


If A + B = 90o, Then


Sin A = Cos B


Sin2A + Sin2B = Cos2A + Cos2B = 1


Tan A = Cot B


Sec A = Csc B


For example:    


If tan (x+y) tan (x-y) = 1, then find tan (2x/3)?


Solution:            


Tan A = Cot B, Tan A*Tan B = 1


So, A +B = 90o


(x+y)+(x-y) = 90o, 2x = 90o , x = 45o


Tan (2x/3) = tan 30o = 1/√3


If A - B = 90o, (A › B) Then


Sin A = Cos B


Cos A = - Sin B


Tan A = - Cot B


If A ± B = 180o, then


Sin A = Sin B


Cos A = - Cos B


If A + B = 180o                   


Then, tan A = - tan B


If A - B = 180o                    


Then, tan A = tan B


 


If A + B + C = 180o, then


Tan A + Tan B +Tan C = Tan A * Tan B *Tan C


sin θ * sin 2θ * sin 4θ = ¼ sin 3θ


cos θ * cos 2θ * cos 4θ = ¼ cos 3θ


For Example:What is the value of cos 20ocos 40o cos 60o cos 80o?


Solution: We know cos θ * cos 2θ * cos 4θ = ¼ cos 3θ


Now, (cos 20o cos 40o cos 80o ) cos 60o


¼ (Cos 3*20) * cos 60o


¼ Cos2 60= ¼ * (½)2 = 1/16


If             a sin θ + b cos θ = m     &    a cos θ - b sin θ = n


then a2 + b2 = m2 + n2


For Example:


If 4 sin θ + 3 cos θ = 2 , then find the value of  4 cos θ - 3 sin θ:


Solution:


Let 2 cos θ - 3 sin θ = x


By using formulae a2 + b2 = m2 + n2


42 + 32 = 22 + x2


16 + 9 = 4 + x2


X = √21


If


sin θ +  cos θ = p     &     csc θ -  sec θ = q


then P – (1/p) = 2/q


For Example:


If sin θ + cos θ = 2 , then find the value of  csc θ - sec θ:


Solution:


By using formulae:


P – (1/p) = 2/q


2-(1/2) = 3/2 = 2/q


Q = 4/3 or csc θ - sec θ = 4/3


If


a cot θ + b csc θ = m     &    a csc θ + b cot θ = n


then b2 - a2  = m2 - n2


If


cot θ + cos θ = x     &    cot θ - cos θ = y


then x2 - y2 = 4 √xy


If


tan θ + sin θ = x     &    tan θ - sin θ = y


then x2 - y2 = 4 √xy


If


y = a2 sin2x + b2 csc2x + c


y = a2 cos2x + b2 sec2x + c


y = a2 tan2x + b2 cot2x + c


then,


ymin = 2ab + c


ymax = not defined


For Example:                    


If y = 9 sin2 x + 16 csc2 x +4 then ymin is:


Solution:            


For, y min = 2* √9 * √16 + 4


= 2*3*4 + 20 = 24 + 4 = 28


If            


y = a sin x + b cos x + c


y = a tan x + b cot x + c


y = a sec x + b csc x + c


then,     ymin = + [√(a2+b2)] + c


ymax = - [√(a2+b2)] + c


For Example:                    


If y = 1/(12sin x + 5 cos x +20) then ymaxis:


Solution:            


For, y max = 1/x min


= 1/- (√122 +52) +20 = 1/(-13+20) = 1/7


Sin2 θ, maxima value = 1, minima value = 0


Cos2 θ, maxima value = 1, minima value = 0




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