Download Period of Tanx, Cotx is π

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M.N. Rao
Senior faculty,
Sri Chaitanya Educational
institutions
TRIGONOMETRY UPTO
INVERSE TRIGONOMETRY
Trigonometric equations & Inverse trigonometric functions are most
important portion of Trigonometry. It
not only contains the concepts of Trigonometric Equations and Identities
but also involves concepts of functions.
The chapter is important not only
because it fetches five to six questions EAMCET examination but also
because it is prerequisite to the other
chapters of Mathematics.
A Trigonometric equations is one
that involves one or more of the six
functions sine, cosine, tangent, cotangent, secant, and cosecant
The trigonometric equation may
have infinite number of solutions.
Solutions are classified as:
1) Principal solution
2) General solution
Principal Solution: Numerically
least angle is called the principal
value.
General solution: The solution
consisting of possible solutions of a
trigonometric equation is called its
general solution.
In most of the case you will find
that one of the following approach
may be employed to proceed further
based on the problem
a) Substitute for one variable say
y in terms of other x i.e. eliminate y and solve the way you
used to solve for trigonometric equations in one variables
b) Extract the linear/algebraic simultaneous equations from
the given trigonometric equations and solve them as algebraic simultaneous equations.
c) Many times you may need to
make appropriate substitutions. It will be particularly useful when the system has only
two trigonometric functions.
Bijective functions only have
inverse functions. Since the trigonometric functions are periodic functions, these functions are not bijections in their natural domains. Therefore, their inverses do not exist.
Important forms of inverse
Trigonometric functions:
1. Problems on simple equations
and in equations involving inverse trigonometric functions.
2. Properties of inverse trigonometric functions with examples.
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Period of Tanx, Cotx is π
3. Domain and range of functions
Five to six questions appeared
in previous year EAMCET in these
chapters
27. If sin–1 x + sin–1 y + sin–1z =
π
2
then x2 + y2 + z2 + 2xyz = 1
28. If cos–1 x + cos–1 y + cos–1 z = π
then x2 + y2 + z2 + 2xyz = 1
27. Period of ax – [ax] is
28. Range of Sinx or Cosx are
[–1, 1]
29. Range of aCosx+bSinx+c is
IMPORTANT FORMULAS
Standard Results
1. Principal value of value θ for
function Sinθ lies between
 −π π 
 2 , 2


 π π
− ,
 2 2
3. Principal value of value θ for function Cosθ lies between [0, π]
4. General solution of Sinθ is nπ +
(–1)n α if Principal value is α
5. General solution of Tanθ is nπ+α
if Principal value is α
6. General solution of Cosθ is 2nπ
± α if Principal value is α
7. General solution of θ if Sin θ = 0
is θ = nπ
8. General solution of θ, if Cos θ =
0 is (2n + 1)
π
2
sinθ = 1 ⇒ θ = (4n + 1)
π
2
π
sinθ = –1 ⇒ θ = (4n − 1)
2
cosθ = 1 ⇒ θ = 2nπ
cosθ = –1 ⇒ θ = (2n+1)π
Most general value for sin θ = sin
α and cos θ = cosα ⇒ θ = 2nπ +
α
11. sin–1(sinθ) = θ if and only if
π
π
− ≤ θ ≤ and
2
2
sin
= x where –1 ≤ x ≤ 1
12. cosec–1 (cosecθ) = θ if and only
(sin–1x)
π
2
if − ≤ θ ≤ 0 or 0 ≤ θ ≤
π
2
and cosec(cosec–1x) = x where
– ∞ < x ≤ –1 or 1 ≤ x ≤ ∞
13. tan–1(tanθ) = θ if and only if
π
π
− <θ<
2
2
and tan (tan–1x) = x where – ∞ < x < ∞
14. cos–1 (cosθ) = θ if and only if
0≤θ<π
and cos (cos–1x) = x where –1 ≤ x ≤ 1
15. sec–1 (secθ) = θ if and only if
π π
0 ≤ θ < or < θ ≤ π
2 2
and sec (sec–1x) = x where
(120°+θ) =
EAMCET
30. Minimum value of a2Sin2x +
b2 Cosec2 x is 2ab
Previous
EAMCET
Questions
3
2
05. Sinθ + Sin (120+θ) – Sin (120–θ)
=0
06. Sinθ.Sin (60–θ).Sin (60+θ)
2015 SPECIAL
x
3
1. Find period of cos + sin
07. cosθ.cos (60o–θ).cos(60o+θ)
– ∞ < x ≤ –1 or 1 ≤ x < ∞
16. cot–1 (cotθ) = θ if and only if
0<θ<π
and cot(cot–1x) = x where –∞<x<∞
π
, tan–1 x +
2
π
, sec–1 x + cosec–1 x
2
cots–1 x =
π
= 2
18. Tan–1x + Tan–1y + Tan–1z =
tan–1
x + y + z − xyz
1 − xy − yz − zx
19. sin–1(3x–4x3) = 3 sin–1x,
cos–1 (4x3 – 3x) = 3cos–1x
20. If 0≤x≤1, 0≤y≤1 cos–1x – cos–1 y
(
−1
2
2
= Cos xy + 1 − x 1 − y
)
21. If 0≤x≤1, 0≤y≤1 sin–1x+sin–1y =
Sin −1 ( x 1 − y 2 + y 1 − x 2 ) for x 2 + y 2 ≤ 1
22.
tan–1 x
–
tan–1 y
=
tan–1
x−y
1 + xy
x≥0
y≥0
tan–1x + tan–1 y =
 −1 x + y
x ≥ 0, y ≥ 0, xy < 1
 tan 1 − xy


−1 x + y
, x > 0, y > 0, xy > 1
π + tan
1 − xy

π
x > 0, y > 0, xy = 1

2
1 − x2
,x ≥0
23. 2 tan–1 x = cos–1
1 + x2
2x
2x
2tan–1x=sin–1 1 + x 2 = tan −1 1 − x2 , x < 1
24. 2 tan–1x π – sin–1
=
2x
2x
= π + tan −1
, x >1
1 + x2
1 − x2
25. If Tan–1x + Tan–1y + Tan–1 z = π
then x + y + z = xyz
26. If Tan–1x + Tan–1y + Tan–1 z =
π
then xy + yz + xz = 1
2
=
x
2
(EAMCET- 2013)
1) 2π 2) 4π 3) 8π 4)12π
Ans: option 4; period
1
= sin 3θ
4
Mathematics
17. sin–1 x + cos–1 x =
9. If Sin2θ = sin2α or Cos2θ = Cos2
α or Tan2θ = Tan2α then general
solution is θ = nπ ± α, where α ∈
Ist Q
10. For a Cosθ + b Sin θ = c then
2
2
solution exists if c ≤ a + b
tanθ = 0 ⇒ θ = nπ
c − a 2 + b2 , c + a2 + b2 


01. Sin4θ + Cos4θ = 1–2Sin2θCos2θ
02. Sin6θ+ Cos6θ = 1–3Sin2θCos2θ
03. Cosθ + Cos (120°–θ) + Cos
(120°+θ) = 0
04. Cos2θ + Cos2 (120°–θ) + Cos2
2. Principal value of value θ for function Tanθ lies between
1
a
x
= 2π , x = 6π,
3
1
cos 3θ
4
x
= 2π , x = 4π
2
08. Tanθ.Tan(60o–θ).Tan(60o+θ)
= Tan3θ
09. Sin (A+B). Sin (A–B)
= Sin2A– Sin2B
10. Cos(A+B).Cos(A–B)
= Cos2A – Sin2B
11. If A + B = 45o or 225o then
(1 + TanA) (1+TanB) = 2;
(1 – CotA) (1–Cot B) = 2
12. If A + B = 135o or 315o then
(1– TanA) (1–TanB) = 2;
(1+CotA) (1+Cot B) = 2
13. If A + B + C = 180o then ΣTanA
= πTanA, ΣCotACotB =1
14. If A + B + C = 90o then
ΣTanA TanB=1, ΣCotA= πCotA
15. CotA+TanA = 2Cosec2A
16. CotA–TanA = 2Cot2A
17. If A±B = 60o then
Cos2A+Cos2B +
thus period LCM = 12π
2. Value of Tan9° – Tan27° – Tan
63° + Tan81° (EAMCET- 2014)
1) 2 2) 3
3) 4 4) 0
Ans: option 3
Tan9° – Tan27° – Tan 63° + Tan
81° = (Tan9° +Cot9°) – (Tan27°
+ Cot27°)
= 2 Cosec18° – 2Cosec54°
3
4
Cos3θ = 1 = Cos2π = Cos4π
which gives 2π/3, 4π/3.
4. If Tan–1x + Tan–1 y + Tan–1z =π then x + y + z = (EAMCET 2014)
1) xyz
2) 3xyz
3) 0
4) xyz
Ans: option 1; formulae.
CosACosB =
18. If A±B = 60o then Sin2A+Sin2B +
SinASinB =
3
4
19. Cos3θ + Cos3 (120o – θ) – Cos3
3
4
(120o + θ) = C os 3θ
20. Tanθ + Tan (60o + θ)
+ Tan (120o + θ) = 3Tan3θ
21. Sin (A+B+C)= ΣSinACosBCosC
– SinASinBSinC
22. Cos(A+B+C)= CosACosBCosC
– Σ SinASinBCosC
23. aCosθ + bSinθ = C
2
2
2
⇒ aSinθ– bCosθ = ± a + b − c
24. Period of Sinx, Cosx, Secx and
Cosecx is 2π
25. Period of Tanx, Cotx is π
26. Period of x – [x] is 1;
4 
 4
= 2
−
=4
5 +1
 5 −1
3. If Cosθ.Cos(60°–θ). Cos(60°+θ)
=
1
then sum of solutions
4
(0≤θ≤4π)
(EAMCET- 2013)
1) 2π 2) 4π 3) 3π 4) π
Ans: option 1
Cosθ.Cos(60°–θ). Cos(60°+θ)
=
5.
1
1
Cos3θ =
4
4
π

π

sin  cot θ  = cos  tan θ  then
4

4

θ = ––––
π
1) nπ +
2
π
3) nπ −
4
(EAMCET 2014)
π
4
π
4) nπ +
3
2) nπ +
Ans: option 2
Substitute n = 0 in option
⇒
π π π π
, ,− ,
2 4 4 3
θ=
π
satisfies given
4
as options