An identity is an equation that always holds true. A trigonometric identity is an identity that contains trigonometric functions and holds true for all right-angled triangles. Sin 2x Cos 2x is one such trigonometric identity that is important to solve a variety of trigonometry questions.
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Sine (sin): Sine function of an angle (theta) is the ratio of the opposite side to the hypotenuse. In other words, sinθ is the opposite side divided by the hypotenuse.
Cosine (cos): Cosine function of an angle (theta) is the ratio of the adjacent side to the hypotenuse. In other words, cosθ is the adjacent side divided by the hypotenuse.
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)
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) |
∫ (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
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)
Example 1: Derive the derivative of sin 2x cos 2x
Solution:
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)
Example 2: Derive the integral of sin 2x cos 2x
Solution:
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.
1. What is Trigonometry? What is a Trigonometric Identity?
Trigonometry is basically the study of the relationships between the sides and angles of a right-angled triangle. This concept was first introduced by the Greek mathematician Hipparchus. The concepts of trigonometry play a key role in physics, music theory, engineering, design, marine biology, and many other diverse fields.
An identity is an equation that always holds true. A trigonometric identity is an identity that contains trigonometric functions and holds true for all right-angled triangles. They are useful when solving questions with trigonometric functions and expressions.
2. What is a Derivative? What is an Integral?
An equation has a variable. Depending on how the variable changes, the final value also changes. The derivative of an expression defines how much the final value changes depending on the change in the variable. Some expressions have more than one variable, for such functions, we define the partial derivatives, each derivative defining the change in the final value with respect to the change in one variable. We can find the derivative through differentiation. The derivative represents the slope of the curve.
The integral of a particular expression is found through integration and it represents the area under the curve. An integral is an antiderivative.
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