8/9/2019 IA-09multiple and Sub Multiple Angles(45-47)

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9. MULT IPLE AND SUBMU LTI PLE ANGLES

Synopsis :

1. i) sin 2A = 2 sin A cos A=

Atan1

Atan22

+

ii) cos 2A = cos2 A sin2 A= 1 2 sin 2A

= 2 cos 2A 1

=Atan1

Atan12

2

+

iii) tan 2A =Atan1

Atan22

2. i) sin 3A = 3 sin A 4 sin3 A

ii) cos 3A = 4 cos3

A 3 cos A

iii) tan 3A =Atan31

AtanAtan32

3

3. The values of Trigonometric functions of some standard angles :

18 36 54 72

sin

4

15

4

5210

4

15 +

4

5210 +

cos

4

5210 +

4

15 +

4

5210

4

15

4. i) sin A . sin (60 A) sin (60 + A) =4

1sin 3A

ii) sin A .sin (120 A) sin (120 + A) =4

1sin 3A

5. i) cos A . cos (60 A) cos (60 + A) =4

1cos 3A

ii) cos A . cos (120 A) cos (120+ A) = 4

1

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

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8/9/2019 IA-09multiple and Sub Multiple Angles(45-47)

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Multiple and Submultiple Angles

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8. i) sin2 + sin2 (60 + ) + sin2 (60 ) = 3/2

ii) cos2 + cos

2(60 + ) + cos

2(60 ) = 3/2

iii) sin2 + sin2 (120 + ) + sin

2 (120 ) = 3/2

iv) cos2 + cos2 (120 + ) + cos

2 (120 ) = 3/2

9. i) If tan A + tan B + tan C = tan A tan B tan C,

then A + B + C = n, n Z.

ii) If tan A tan B + tan B tan C + tan C tan A = 1,

then A + B + C = (2n + 1) /2, n Z.

10. cosx cos2x cos4x cos (2nx) =xsin2

)x2sin(1n

1n

+

+

.

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