COMPOUND ANGLES
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FORMULAE
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1) sin(A+B)= sinA cosB+ cosA sinB
2) sin(A--B)= sinA cosB- cosA sinB
3) sin(A+B+C)= sinA cosB cosC+ sinB cosA cosC+ sinC cosA cosB + sinA sinB sinC
= ΣsinA cosB cosC - Π sinA
4) cos(A+B)= cosA cosB-sinA sinB
5) cos(A-B)= cosA cosB+ sinA sinB
6) cos(A+B+C)= cosA cosB cosC -sinA sinB cosC - sinA cosB sinC - cosA sinB sinC.
= Π cosA - ΣsinA sinB cosC
= CosA cosB cosC[1- tanA tanB - tanB tanC - tanC tanA]
7) tan(A+B)= (tanA+tan B)/(1- tanA tanB)
8) tan(A-B)= (tanA-tan B)/(1+ tanA tanB)
9) tan(A+B+C)= (tanA+tan B+ tanC - tanA tanB tanC)/(1- tanA tanB - tanB tanC - tanC tanA)
= (S₁ - S₃)/(1- S₂)
10) cot(A+B)= (cotA cot B-1)/(cotB + cotA)
11) cot(A-B)= (cotA cot B+1)/(cotB - cotA).
12) sin(A+B).Sin(A-B)= sin²A- sin²B
= cos²B - cos²A
13) cos(A+B)cos(A-B)=cos²A- sin²B
= cos²A - sin²B = c
MAXIMUM & MINIMUM VALUES OF TRIGONOMETRIC EXPRESSIONS:
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1) a cosx + b sinx will always lie in the interval [- √(a²+b²), √(a²+b²)] i.e. the maximum and minimum values are √(a²+b²), - √(a²+b²)
2) Minimum value of a² tan²x+ b² cot²x= 2ab, where a,b > 0.
3) - √(a²+b²+ 2ab cos(x-y) ≤ a cos(x+m) + b cos(y+m) ≤ √(a²+b²+ 2ab cos (x-y) where x, y are known angles.
4) Minimum value of a² cos²x + b² sec²x is either 2ab or a²+ b², if for some real x equation a cosx= b secx is true or not true. (a, b > 0)
5) Minimum value a² sin²x + b²cosec²x is either 2ab or a²+ b², if for some real x equation a sinx= b cosecx is true or not true (a,b > 0)
EXERCISE -1
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1) Find the expansion of
a) sin(A+B-C)
b) cos(A-B-C)
c) tan(A-B+C)
d) cot(A+B+C)
e) sec(A-B+C)
f) cosec (A- B- C)
EXERCISE-2
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1) Find the values of
a) sin75° (√3+1)/2√2
b) cos105° -(√3-1)/2√2
c) cot 15° 2+√3
d) cosec(-105°) √2-√6
e) sec (-75°) √2+√6
2) PROVE:
a) cos15° - sin15°= 1/√2
b) sin15°+ sin75°= √(3/2).
c) tan75°- cot 75°= 4 sin60°
d) sin15°+ tan 30°cos15°= √6/3.
e) sin165° + cos165°= -1/√2
f) sin105° + cos105°= cos45°
g) cos80 cos 20 + sin 80 sin 20=½
h) cos84º25′ cos24º25′ + cos5º35′ cos65º35 ′= ½
i) cos80°40'. Cos39°20' - sin80°40'. sin39°20'= -1/2.
j) cos(30°-x) cos(60°-y)- sin(30°-x) sin(60°-y)= sin(x+y).
k) cos(45- x)cos(45-y) - sin(45-x) sin(45 -y) = sin (x+y)
l) cos(π/4 - x) cos(π/4 -y)- sin(π/4 - x) sin(π/4 -y)= sin(x+y)
m) sin(A+B)sin(A- B) + cos(A+B) cos(A -B) + 2 sin²(B) = 1.
n) sin(n+1)A. cos(n-1)A - cos(n-1)A. sin(n-1)A = sin2A.
o) cosA sin(B-C) + cosB sin(C-A) + cos C sin(A -B) =0.
p) sin(A+B). sin(A-B) +sin(B+C). sin(B-C) +sin(C+A) sin(C-A) = 0.
q) sinA sin(B- C)+ sinB sin(C-A)+ sinC sin(A-B)= 0.
r) cosA sin(B-C)+ cosB sin(C-A)+ cosC sin(A-B)= 0.
s) sin(B+C) sin(B-C)+ sin(C+A) sin(C-A) + sin(A+B) sin(A-B)= 0.
t) sin(A- B) sin(C- D) + sin(C-B) sin(D- A) + sin(D- B) sin(A- C)= 0.
u) sin2x cosx + cos2x sinx = sin4x cosx - cos4x sinx.
v) cos(A- B)- sin(A+B)= (cosA - sinA)(cosB - sinB).
w) cos(A+ B)+ sin(A- B)= (cosA + sinA)(cosB - sinB).
EXERCISE --3
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PROVE:
a) sec(x-y)=(secx secy)/(1+tanx tany)
b) cosec(x+y) = (cosec x cosec y)/(coty + cot x).
c) sec(x+ y)=(secx secy)/(1- tanx tany)
d) sin(B- C)/(cosB cosC) + sin(C- A)/(cosC. cos A) + sin(A- B)/(cosA cos B) = 0.
e) sin(B- C)/(sinB. sinC) +sin(C- A)/(sinC. sin A) +sin(A- B)/(sinA. sin B) = 0.
f) tan 20+tan 25 + tan 20 tan25 =1
g) tan 3x - tan2x - tanx =tan3x tan2x tanx.
h) tan(y-z) +tan(z-x)+tan(x-y)= tan(y-z) tan(z-x) tan (x-y).
i) tan(B+C-2A)+ tan(C+A-2B)+ tan(A+B- 2C)= tan(B+C-2A) tan(C+ A - 2B) tan(A+B- 2C).
j) √3(tan 170 - tan 140) =1+tan 170 tan 140.
k) tan 75 - tan30 - tan 75 tan30= 1
l) If tan10+ tanx= 1- tan10 tanx find the value of x. 35°
m) (cosA+sinA)/(cosA- sinA) =tan(π/4 + A).
n) (cos7+sin7)/(cos7- sin7)= tan52
o) (cos10- sin10)/(cos10+sin10)= tan35
p) (cos15+sin15)/(cos15-sin15)= √3
q) (cos11+sin11)/(cos11-sin11)= tan56.
r) tan(45°+A)= (cosA+ sinA)/(cosA- sinA).
s) √2 sin(45+A)= sinA+ cosA.
t) (cos8+ sin8)/(cos8 - sin8)=tan53
u) tan(45+A) tan(45-A)= 1.
v) cot(45+x)= (cotx -1)/(cotx+1) = (cosx- sinx)/(cosx+ sinx).
EXERCISE-4
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PROVE:
a) 1+tan2α. tanα =sec2α
b) cos(A-B)/2 - sinA. sin (A+B)/2 = cosA. cos (A+B)/2
c) cot θ - cot 2θ = cosec2θ.
d) cosA+ cos(120+A)+ cos(120-A)= 0.
e) tan(x+y) + tan(x-y)= sin2x/(1- sin²x - sin²y).
f) tan(x+y) + tan(x-y)= sin2x/(cos²x - sin²y).
g) cot(A+B) + cot(A-B) = Sin2A/(sin²A - sin²B)
h) tan²A - tan²B = {sin(A+B) sin(A-B)}/cos²A cos²B.
i) cotA/(cotA- cot3A) - tanA/(tan3A - tanA)= 1
j) 1/(tan3A - tanA) - 1/(cot3A - cotA) = cot2A.
k) tan(A+B) tan(A-B)= (sin²A- sin²B)/(cos²A - cos²B).
l) {tan(A+B)- tanA}/{1+ tan(A+B) tanA)}= tan B.
m) sin(x+y)}/sin(x-y)= (tanx+ tany)/(tanx - tany).
m) sinx + cosx =√2 sin(π/4 +x)= √2 cos (π/4 - x).
n) cos 15 + √3 sin15=√2
o) {sin(2A+B)}/sinA - 2 cos (A+B)= sinB/sinA
EXERCISE-5
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1) Given sinα=4/5, sin β=3/5 find cos(α+β). 0
2) Given sinα=3/5, tan β=5/12 find sin(α+β). 56/65
3) Given secα=5/4, cosec β=13/12 find sec(α - β). 65/56
4) If sinα= 1/√10, cos β=2/√5 find the value of (α+β). π/4
5) Show that Sin(A-B)=16/65, cos(A+B)= 33/65 if sinA= 2/5, cosB= 12/13.
6) sec(A-B)=85/84, if secA= 17/8, cosecB= 5/4.
7) If sinx= 3/5, sin y= 8/17, find
A) sin(x± y). 77/85,13/85
B) cos(x± y). 36/85, 84/85
C) tan(x± y) 77/36, 13/84
8) if sinx= 3/5, cos y= 5/13, x and y in Quadrant I. Then find the value of
A) sin(x+y). 63/65
B) cos(x+y). -16/65
C) tan(x+y). -63/16
9) if cosx= -12/13, cot y= 24/7, x in II and y in Quadrant III. Then find the value of
A) sin(x+y). -36/325
B) cos(x+y). 323/325
C) tan(x+y). -36/323
10) if sinx= 8/17, tan y= 5/12, x and y in Quadrant I. Then find the value of
A) sin(x-y). 21/221
B) cos(x-y). 220/221
C) tan(x-y). 21/220
11) if sinx= -12/13, cot y= 24/7, x in I quadrant and y in Quadrant III. Then find the value of
A) sin(x-y). 204/325
B) cos(x-y). -253/325
C) tan(x-y). -204/253
12) If tanA= 5/6, tanB= 1/11 then find the value of (A+B). 45°
13) If tanA= m/(m+1), tanB= 1/(2m+1) then find the value of (A+B). π/4
14) A, B, C are three positive acute angles and tanA= 1/2, tanB= 1/5, tanC= 1/8, show that A+B+C= π/4.
15) sin (x+y)= 4/5, cos(x-y)= 12/13 and 0< x< π/4, 0< y< π/4; find the value of tan 2x. 33/56
EXERCISE --6
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PROVE:
1) If 2 tan B + cot B= tan A, show that cotB= 2 tan(A- B).
2) cos²A + cos²(120°+A)+cos²(120-A) = 3/2.
3) If tanA= 2 tanB, find the value of {sin(A+B)}/{sin(A-B)}. 3
4) If tan x - tan y =a, cot y - cot x=b Show that1/a + 1/b = cot(x - y)
5) If tan3A- tanA= x and cot3A - cotA= y, show cot2A= 1/x - 1/y.
6) If sin(A+B)= n sin(A-B), then show that tanA= {(n+1)/(n-1)} tanB.
7) If tan(A+B)= n tan(A-B) then show (n+1) sin2B= (n-1) sin2A.
8) If m tan(A-30)= n tan(A+120), then show cos2A=(m+n)/{2(m-n)}.
9) If tan m= (K sin a. cos a)/1- Ksin² a), show tan(a-m)= (1-K) tan a.
10) If sin a= A sin(a+b), show that tan(a+b)= sin b/(cos b - A).
11) If a cos(m+n)= b cos(m -n), show that (a+b) tan m= (a-b) cot n.
12) If tan a= x sin b/(1- x cos b) and tan b= y sin a/(1- y cos a) prove that sin a: sin b= x: y.
13) if cos (A-B) sin(C- D)= cos(A+B) sin(C+D), show that tan D= tan A tanB tan C.
14) If sin a sin b - cos a cos b +1= 0, then show that 1 + cot a tan b= 0.
15) If tan m= (asinx + bsiny)/(a cosx-+ b cosy), then show that a sin(m -x) + b sin(m-y)= 0.
16) If cotA= cos(x+y) and cotB= cos(x-y), then show tan(A-B)=(2sinx siny)/(cos²x + cos²y).
17) An angle a is divided into two parts m and n such that sim m: sin n= K: 1, show that tan m= K sin a/(1+ K cos a) and tan m= sin a/(K+ cos a).
18) An angle m is divided into two parts, a, b such that tan a: tan b= x: y; prove sin(a-b)= {(x-y)(sin m}/(x+y)
19) If A+ B= π/4, show that (1+ tanA)(1+tanB)= 2.
20) If A+ B= 45°, show that tan A+ tanB + tanA tanB= 1.
Hence otherwise, express tan22°30' in surd form. √2 - 1.
21) Given that A= B+ C, show that tanA - tanB - tan C=tanA tanB tanC.
22) If A+ B+C= 2π and sinA= sinB cosC, prove 2tanB + tanC= 0.
23) If A+ B= 135, show that (1+ tanA)(1+ tanB)= 2 tanA tanB.
24) If A, B, C are in AP and B=π/4; what is the value of tanA tanB tanC. 1
25) If tan y= 2 sin z sin x cosec(z+x), show that cotx, coty, cotz are in AP.
26) If A+ B+C=π, Prove that,
A) tanA+ tanB+ tanC= tanA tanB tanC.
B) tanA/2 tanB/2+ tanB/2 tanC/2 + tanC/2 tanA/2= 1.
C) cotA/2 + cotB/2+ cotC/2= tanA/2 + cotB/2 cotC/2.
27) If A+ B+ C=π/2, show that cotA+ cotB+ cotC= cotA cotB cotC.
28) If A+ B= 225°, show tanA+ tanB= 1 - tanA tan B.
29) In a ∆ ABC If cosA= cosB cosC, prove that
A) tanA= tanB+ tan C
B) cotB. cotC= 1/2
30) If cos(B-C)+ cos(C-A)+ cos(A-B) =-3/2, then show that ∑ cos a= 0 and ∑sin a = 0.
31) If cosx + cosy+ cos z=0 and sin x + sin y+ sin z=0, show that,
A) cos(x-y)= cos(y-z)= cos(z-x)=-1/2
B) cos(x-y)+ cos(y-z)+cos(z-x)=-3/2
32) Express a cosx + b sinx in the form of r sin (θ+α) or r cis(θ -α) from which find the values of r, α,θ
33) Find Maxumum and minimum value of 5 cos x + 12 sin x -12.
34) Find Maxumum and minimum value of 3cos x+ 4sin x +5. 10, 0
35) Eliminate A and B:
A) tanA+ tanB= a, cotA+ cotB= b, A+B= k
B) sinA+ sinB= a, cosA+ cosB= b, A-B= π/3.
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