[Maths Class Notes] on Sin 2x Cos 2x Pdf for Exam

Derivation of Sin 2x Cos 2x

We make use of the trigonometry double angle formulas, to derive this identity:

We know that, (sin 2x = 2 sin x cos x)————(i)

cos 2x = cos2 x − sin2 x

= 2 cos2 x − 1 [because sin2x + cos2 x = 1]——(ii)

= 1 − 2 Sin2x——————————————-(iii)

We want to find the value of sin 2x cos 2x. To do this, multiply equation (i) and (ii).

Sin 2x = 2 sin x cos x

Cos 2x = 2 cos2x − 1

Multiply the above two answers to get the value:

sin 2x cos 2x = (2 sin x cos x) (2 cos2x − 1)

= 2 cos x (2 sin x cos2 x − sin x)

Now, consider equation (i) and (iii),

sin 2x = 2 sin x cos x

cos 2x = 1 − 2 sin2x

Multiply them to get,

sin 2x cos 2x = 2 sin x cos x (1 − 2 Sin2x)

= 2 cos x (sin x – 2 sin3 x)

Value of Sin 2x Cos 2x

Sin 2x Cos 2x = 2 Cos x (2 Sin x Cos2 x − Sin x)

(or)

Sin 2x Cos 2x = 2 Cos x (Sin x – 2 Sin3 x)

Integral of Sin 2x Cos 2x

∫ (Sin 2x Cos 2x) = (Sin 2x)2/ 4 + C

Proof:

Consider sin 2x = y

Then dy/dx = 2 cos 2x (or) dx = dy / 2 cos 2x

Now, ∫y cos 2x dx = ∫y • cos(2x) • dy / 2 cos 2x

Cancel out cos 2x.

∫y Cos(2x)dx = ∫(y • dy/2)

= ½ [ ∫y dy ]

= ½ y²/2 + c

= y²/4 + C

Therefore, the integral of sin 2x cos 2x is ∫ (Sin 2x Cos 2x) = (Sin 2x) 2 / 4 + C

Derivative of Sin 2x Cos 2x

d/dx (Sin 2x Cos 2x) = 2Cos(4x)

Proof:

Sin 2x cos 2x = ½ (2 sin 2x cos 2x) (Or) ½ sin 4x

By differentiating the given function:

d/dx [ ½ sin 4x ] = ½ [d/dx (sin 4x)]

= ½ [cos 4x d/dx(4x) ]

= ½ [cos (4x) (4) ]

Therefore, the derivative of sin 2x cos 2x is d/dx (Sin 2x Cos 2x) = 2 Cos (4x)

Solved Examples

Example 1: Derive the derivative of sin 2x cos 2x

Solution: