Double angle formulas involving sin and cosine :
Problem 1 :
The expression 2sin 30° cos 30° has the same value as
a. sin 15° b. cos 60° c. sin 60° d. cos 15°
Solution:
= 2sin 30° cos 30°
= 2 × (1/2) × (√3/2)
= √3/2
= sin 60°
So, option (C) is correct.
Problem 2 :
The expression csc A sin 2A is equivalent to
a. 2sin A b. 2 c. 2cos A d. 2cot A
Solution:
= csc A sin 2A
So, option (C) is correct.
Problem 3 :
If sin A = 2/3, find cos 2A.
Solution:
Problem 4 :
If sin A = 3/5, find cos 2A.
Solution:
Problem 5 :
If θ is in Quadrant II and cos θ = -3/4, find an exact value for sin 2θ.
Solution:
Problem 6 :
Solution:
Verify each identity.
Problem 7 :
2sin x cos x - cos 2x = 2sin2x - 1 + sin 2x
Solution:
2sin x cos x = sin 2x
cos 2x = 1 - 2sin2x
2sin x cos x - cos 2x = 2sin2x - 1 + sin 2x
Hence, it is verified.
Problem 8 :
cos2x + tan2x sin2x = sec2x sin2x + cos2x
Solution:
R.H.S :
= sec2x sin2x + cos2x
sec2x = 1 + tan2x
= (1 + tan2x) sin2x + 1 - 2sin2x
= sin2x + tan2x sin2x + 1 - 2sin2x
= tan2x sin2x + 1 - sin2x
1 - sin2x = cos2x
= tan2x sin2x + cos2x
L.H.S
Problem 9 :
Solution:
Hence, it is verified.
Problem 10 :
1 + cos 2x - tan 4x cos 4x = 2cos2x - sin 4x
Solution:
L.H.S :
= 1 + cos 2x - tan 4x cos 4x ----(1)
cos 2x = 2 cos2 x - 1
1 + cos 2x = 2 cos2 x
tan 4x = sin 4x / cos 4x
Applying these values in (1), we get
= 1 + cos 2x - (sin 4x / cos 4x) cos 4x
= 1 + cos 2x - sin 4x
= 2cos2x - sin 4x
R.H.S
Problem 11 :
Solution:
Hence, it is verified.
Problem 12 :
Solution :
Problem 13 :
Solution :
Problem 14 :
Solution :
Problem 15 :
Solution :
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May 21, 24 08:51 AM
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