Examples on Derivative of sin 2x

Example 1 : If cos A = 3/5 where A is in quadrant I, then find the value of sin 2A?

Solution:

According to Pythagorean identity

sin2A + cos2A = 1

sin2A = 1 – cos2A

sin A = ±√(1 − cos2A)

sin A = ±√(1 − (3/5)2)

sin A = ±√(16/25)

sin A = ± 4/5

Since A is in quadrant I, sin⁡ A is positive. Thus,

sin A = 4/5

From sin 2x formula, sin 2x = 2 sin x cos x. From this,

sin 2A = 2 sin A cos A

= 2 (4/5) (3/5)

= 24/25

Example 2: Find the value of sin 90 Degrees using the Sin2x Formula.

Solution:

Sin2x Formula is given as Sin (2x) = 2Sin x Cos x. To find the value of sin 90 degrees, Sin2x Formula can to be used.

2x = 90^o

x = 90°/2

x = 45°

Value of x is obtained. Substituting its value into the Sin2x Formula,

Sin (2 x 45°) = 2sin45° cos45°

We know that sin45° = 1/√2 and cos 45° = 1/√2. Using these values we get,

Sin 90°=2×1/√2 x 1/√2

Sin 90° = 1

Example 3: Determine the value of sin 2x if sin x = 4/5

Solution:

Given that, sin x = 4/5, Using the Pythagorean theorem we can obtain that, cos x = 3/5.

According to Sin2x formula we get,

sin 2x = 2 sin x cos x

Putting the given sin x value and cos x value, we get

sin 2x = 2 (4/5) (3/5)

sin 2x = 24/25

Example 4: Calculate the derivative of sin(2x+1)?

Solution:

Let f(x) = sin(2x+1)

Applying Chain rule,

⇒f'(x) = cos(2x+1) d/dx(2x+1)

⇒cos(2x+1)(2)

⇒2 cos(2x+1)

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.