www.sakshieducation.com
COMPOUND ANGLES
SYNOPSIS
1.
sin(A+B) = sinAcosB + cosAsinB
sin(A-B) = sinAcosB - cosAsinB
2.
cos(A+B) = cosAcosB - sinA sinB
cos(A-B) = cosAcosB + sinAsinB
3.
4.
5.
tan(A+B) =
tan A + tan B
1 − tan A tan B
tan(A-B) =
tan A − tan B
1 + tan A tan B
cot(A+B) =
cot A cot B − 1
cot A + cot B
cot(A-B) =
cot A cot B + 1
cot B − cot A
sin(A+B) sin(A-B) = sin2A-sin2B = cos2 B -cos2A
cos(A+B) cos(A-B) = cos2A - sin2B = cos2 B - sin2 A
6.
3
4
If A ± B = 60 0 , then cos 2 A + cos 2 B − cos A cos B = ,
If A ± B = 60 0 , then sin 2 A + sin 2 B ± sin A sin B = 3 / 4
7.
Sin 2 A + sin 2 B = 1 − cos(A + B) cos(A − B)
Cos 2 A + cos 2 B = 1 + cos(A + B) cos(A − B)
8.
sin 150 =
3 −1
3 +1
, cos 150 =
, tan 150 = 2- 3
2 2
2 2
9.
sin 750 =
3 +1
3 −1
, cos 750 =
, tan 750 = 2+
2 2
2 2
10.
If A+B+C = 1800 , then
i)
tanA + tanB + tanC = tanA tanB tanC
ii)
tan2A + tan2B + tan2C = tan2A tan2B tan2C
3
iii) tan3A + tan3B + tan3C = tan 3A tan3B tan3C
iv)
tan
A
B
B
C
C
A
tan + tan tan + tan tan = 1
2
2
2
2
2
2
www.sakshieducation.com
www.sakshieducation.com
v)
cotA cotB + cotB cot C + cot C cotA = 1
vi)
cot
A
B
C
+ cot + cot = cot A / 2 cot B / 2 cot c / 2
2
2
2
11.
If A = B+C, then tanA - tanB-tanC = tanA tanB tanC.
12.
If A, B, C, D are the angles of cyclic quadrilateral, then cosA+cosB+cosC+cosD = 0
13.
The range of the expression a cosθ + b sinθ is
14.
The maximum value of a cosθ + b sinθ is
[−
a2 + b2
a2 + b2 , a2 + b2
]
and its minimum value is -
15.
3π
c+s > 0
π
Y
4
4
c-s<0
c+s<0
X1
c+s > 0
c-s<0
c-s>0
X
Here c + s is
cosθ + sin θ
c+s < 0
c-s >0
−3π
or
5π
4
16.
(i)
If
−π
Y1
4
4
or
7π
4
−π
π
< θ < . cosθ + sinθ > 0, cosθ - sinθ > 0
4
4
(ii) If
π
3π
<θ<
cosθ + sinθ > 0, cosθ - sinθ < 0
4
4
(iii) If
3π
5π
<θ<
cosθ + sinθ < 0, cosθ - sinθ < 0
4
4
(iv) If
5π
7π
<θ<
cosθ + sinθ < 0, cosθ - sinθ > 0
4
4
m+ n
m− n
, tan2B =
1 − mn
1 + mn
17.
If tan(A+B)=m, tan(A-B)=n, then tan2A =
18.
cos θ + sin θ
cos θ − sin θ
= tan ( 45 0 + θ) ,
= tan ( 45 0 − θ)
cos θ − sin θ
cos θ + sin θ
19.
a)
20.
cot A + cot B sin( B + A)
=
.
cot A − cot B sin( B − A)
21.
a) sin(A+B+C) = cosAcosBcosC (tanA+tanB+tanC – tanA tan B tan C)
tan A + tan B sin( A + B)
=
tan A − tan B sin( A − B)
b) If
p + q sin(α + β )
tan α p
= then
=
tan β q
p − q sin((α − β )
b) cos(A+B+C) = cosAcosBcosC (1-tanA tanB – tanB tanC – tanC tanA)
c) tan ( A + B + C) =
tan A + tan B + tan C − tan A tan B tan C
1 − tan A tan B − tan B tan C − tan C tan A
www.sakshieducation.com
a2 + b2
.
www.sakshieducation.com
22.
Tan (A1 + A 2 + A 3 + A 4 + − − − − ) =
S1 − S3 + S5 − − −
, where S n = sum of the products of
1 − S 2 + S 4 − S6 + − −
the tangents taken 'n' at a time.
23.
If, in triangle ABC, cotA + cotB + cotC = 3 , then the triangle is equilateral.
24.
If A + B= , then
π
4
i) (1 + tan A )(1 + tan B) = 2
ii) (1 − cot A )(1 − cot B) = 2
25.
If A+B =
3π
, then
4
i) (1 − tan A )(1 − tan B) = 2
ii) (1 + cot A )(1 + cot B) = 2
26.
If A - B =
3π
, then
4
i) (1 − tan A )(1 + tan B) = 2
ii) (1 + cot A )(1 − cot B) = 2
27.
(a) cos2 A + cos2 (600 -A) + cos2 (600 + A) = 3/2
(b) sin2 A + sin2 (600 -A) + sin2 (60 + A) = 3/2
(c) cos2A + cos2 (1200 + A) + cos2(1200 - A)= 3/2
(d) sin2A+sin2(1200 +A) + sin2 (1200 - A) = 3/2
28.
a) In a triangle ABC, the minimum value of tan2
A
B
C
+ tan 2 + tan 2
is unity.
2
2
2
b) In a triangle ABC, the minimum value of tanA + tanB + tanC is 3 3
29.
If α, β are solutions of the equation a tanθ + b secθ = c
2ac
c2 − b 2
2
2
then tanα + tan β = 2 2 , tanα tanβ = 2
and tan (α+β) = a − c
2
a −b
a −b
2ac
a 2 + b2 − 2
2
30.
If cosx + cosy = a, sinx + siny=b, then cos(x-y) =
31.
cot 60 0 + θ cot 60 0 − θ + cot θ cot 60 0 − θ -cot 60 0 + θ cot θ = 3
(
) (
)
(
)
(
)
www.sakshieducation.com