DOUBLE ANGLE TRIG IDENTITIES PRACTICE PROBLEMS

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 :