Download Multiple and Sub Multiple Angles

9. MULTIPLE AND SUBMULTIPLE ANGLES
Synopsis :
1. i) sin 2A = 2 sin A cos A
=
2 tan A
1 + tan2 A
ii) cos 2A
= cos2 A – sin2 A
= 1 – 2 sin 2A
= 2 cos 2A – 1
=
iii)tan 2A =
2.
1 + tan2 A
2 tan A
1 − tan 2 A
i) sin 3A = 3 sin A – 4 sin3 A
ii) cos 3A = 4 cos3 A – 3 cos A
iii)tan 3A =
3.
1 − tan2 A
3 tan A − tan3 A
1 − 3 tan2 A
The values of Trigonometric functions of some standard angles :
θ
sin
18°
5 −1
36°
5 +1
10 + 2 5
5.
i) sin A . sin (60° – A) sin (60° + A) =
4
4
10 − 2 5
4
4
4.
5 +1
10 − 2 5
4
cos
54°
4
1
sin 3A
4
ii) sin A .sin (120° – A) sin (120° + A) =
1
sin 3A
4
i) cos A . cos (60° – A) cos (60° + A) =
1
cos 3A
4
1
4
ii) cos A . cos (120° – A) cos (120°+ A) = cos 3A
6.
i) tan A .tan (60° – A) tan (60° + A) = tan 3A
ii) tan A .tan (120° – A) tan (120° + A) = tan 3A
7.
i) cot A cot (60° – A) cot (60° + A) = cot 3A
ii) cot A cot (120° – A) cot (120° + A) = cot 3A
1
72°
10 + 2 5
4
5 −1
4
Multiple and Submultiple Angles
8.
i) sin2 θ + sin2 (60° + θ) + sin2 (60° – θ) = 3/2
ii) cos2 θ + cos2 (60° + θ) + cos2 (60° – θ) = 3/2
iii)sin2 θ + sin2 (120° + θ) + sin2 (120° – θ) = 3/2
iv) cos2 θ + cos2 (120° + θ) + cos2 (120° – θ) = 3/2
9.
i) If tan A + tan B + tan C = tan A tan B tan C,
then A + B + C = nπ, n ∈ Z.
ii) If tan A tan B + tan B tan C + tan C tan A = 1,
then A + B + C = (2n + 1) π/2, n ∈ Z.
10. cosx cos2x cos4x … cos (2nx) =
sin(2n +1 x )
2n +1 sin x
.
2