What is the Derivative of Sin 2x
Derivative of sin 2x refers to the rate of change of sin 2x to the independent variable x. Sin 2x is a trigonometric function in which the angle of the sine function is expressed as twice the angle. Using the double angle formula in trigonometry, we can find the sine of the angle whose value is doubled. The most commonly used formula of sin 2x is twice the product of the sine function and cosine function, which is mathematically given by, sin 2x = 2 sin x cos x.
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Derivative of Sin 2x Formula
Derivative of sin 2x is given as
d/dx(sin 2x) = 2.cos 2x
Sin 2x Formula Proof
The formula for sin 2x can be derived by using the angle sum formula for sine.
Let us see the derivation of sin 2x step by step:
We know that the sum formula of sin is sin(A + B) = sin A cos B + sin B cos A.
Substitute A = B = x in the formula sin(A + B) = sin A cos B + sin B cos A, we get
sin(x + x) = sin x cos x + sin x cos x
⇒ sin 2x = 2 sin x cos x
Hence, we have derived the formula of sin 2x.
Sin 2x derivative in terms of Tan
We can also write the formula of sin 2x in terms of tan or tangent function only.
We already know the sin 2x formula, sin 2x = 2 sin x cos x
Multiplying and dividing the above equation by cos x, we get
sin 2x = (2 sin x cos2x)/(cos x)
= 2 (sin x/cosx ) × (cos2x)
We know that sin x/cos x = tan x and cos x = 1/(sec x). So
sin 2x = 2 tan x × (1/sec2x)
Using Pythagorean trigonometric identities, sec2x = 1 + tan2x. Substituting this, we have
sin 2x = (2tan x)/(1 + tan2x)
Therefore, the sin 2x formula in terms of tan is sin 2x = (2tan x)/(1 + tan2x).
Derivative of Sin 2x
Derivative of sin 2x is 2cos 2x. Sin 2x is a trigonometric function in which the angle of sin is represented as twice an angle. The trigonometric expansion of sin 2x is 2sinxcosx. The derivative of sin 2x is the rate of change in the function sin 2x to the independent variable x.
In this article, we will learn what is derivative of sin 2x is and how to differentiate sin 2x using various methods in calculus.