How to Find the Value of Sin 2x Cos 2x Easily
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
Integral of Sin 2x Cos 2x
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
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:
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.
FAQs on Sin 2x Cos 2x: Formula, Calculation & Shortcuts
1. What are the fundamental formulas for sin(2x) and cos(2x) in trigonometry?
The fundamental double angle formulas for sin(2x) and cos(2x) are essential identities derived from the sum formulas. The primary formulas you need to know are:
For sin(2x): The most common formula is sin(2x) = 2 sin(x) cos(x).
For cos(2x): This identity has three main forms:
cos(2x) = cos²(x) - sin²(x)
cos(2x) = 2cos²(x) - 1 (by substituting sin²(x) = 1 - cos²(x))
cos(2x) = 1 - 2sin²(x) (by substituting cos²(x) = 1 - sin²(x))
In terms of tan(x), the formulas are sin(2x) = 2tan(x) / (1 + tan²(x)) and cos(2x) = (1 - tan²(x)) / (1 + tan²(x)).
2. How are the double angle formulas for sin(2x) and cos(2x) derived?
The double angle formulas are not arbitrary rules; they are derived directly from the angle sum identities, which are a cornerstone of trigonometry. The process is straightforward:
Derivation of sin(2x): We start with the sum formula sin(A + B) = sin(A)cos(B) + cos(A)sin(B). By setting A = x and B = x, we get sin(x + x) = sin(x)cos(x) + cos(x)sin(x), which simplifies to sin(2x) = 2sin(x)cos(x).
Derivation of cos(2x): We use the sum formula cos(A + B) = cos(A)cos(B) - sin(A)sin(B). By setting A = x and B = x, we get cos(x + x) = cos(x)cos(x) - sin(x)sin(x), which gives us the first form: cos(2x) = cos²(x) - sin²(x). The other two forms are then derived by substituting the Pythagorean identity (sin²(x) + cos²(x) = 1).
3. What is the key difference between an expression like cos(2x) and 2cos(x)?
This is a common point of confusion. The two expressions represent fundamentally different transformations of the basic cosine function:
cos(2x) involves a change in the angle. The '2' inside the function doubles the frequency of the cosine wave, meaning it completes its cycle twice as fast as cos(x). This is a horizontal compression of the graph.
2cos(x) involves a change in the amplitude. The '2' outside the function doubles the peak height and depth of the cosine wave. The range becomes [-2, 2] instead of [-1, 1]. This is a vertical stretch of the graph.
For example, if x = 30°, cos(2x) = cos(60°) = 0.5, whereas 2cos(x) = 2cos(30°) = 2 * (√3/2) = √3 ≈ 1.732. They yield very different values.
4. Why does cos(2x) have multiple formulas, and how do you decide which one to use?
The existence of three different forms for cos(2x) provides great flexibility in solving problems. The choice of which formula to use is strategic and depends on the goal:
Use cos(2x) = cos²(x) - sin²(x) when the problem already involves both sin²(x) and cos²(x), as it allows for direct substitution or factorization.
Use cos(2x) = 2cos²(x) - 1 when you want to convert an expression entirely into terms of cosine. This is useful for solving equations where only cos(x) is present.
Use cos(2x) = 1 - 2sin²(x) when you want to convert an expression entirely into terms of sine. This is ideal for solving equations where you want to eliminate the cosine term.
Choosing the right form simplifies the problem significantly, often reducing a complex equation to a simple quadratic in terms of sin(x) or cos(x).
5. In what types of mathematical problems are the sin(2x) and cos(2x) formulas most useful?
These double angle formulas are not just for theoretical proofs; they are powerful tools used in various practical scenarios, including:
Solving Trigonometric Equations: They help simplify equations containing a mix of angles, like solving for x in an equation such as sin(2x) - sin(x) = 0.
Simplifying Expressions: They are used to reduce complex trigonometric expressions into simpler forms, for example, simplifying (1 - cos(2x)) / sin(2x) to tan(x).
Calculus (Integration and Differentiation): They are crucial for rewriting functions into a form that is easier to integrate or differentiate. For instance, integrating sin²(x) is difficult, but integrating its equivalent form (1 - cos(2x))/2 is straightforward.
Physics and Engineering: These formulas appear in wave mechanics, optics, and electrical engineering to describe wave interference and oscillations.
6. How do double angle formulas help in solving complex calculus problems like integration?
Double angle formulas are a key technique for simplifying integrands in calculus. Powers of sine and cosine, like ∫sin²(x)dx or ∫cos⁴(x)dx, cannot be integrated directly. The strategy is to use the cos(2x) formulas to reduce the power of the function:
To integrate sin²(x), we first rearrange the formula cos(2x) = 1 - 2sin²(x) to get sin²(x) = (1 - cos(2x))/2. The integral of (1/2) - (cos(2x)/2) is simple to compute.
Similarly, to integrate cos²(x), we rearrange cos(2x) = 2cos²(x) - 1 to get cos²(x) = (1 + cos(2x))/2, which is also easy to integrate.
This method, known as 'power reduction', transforms a difficult problem into a standard integral, making it a vital tool for students studying calculus.
7. How can you simplify the product of sin(x) and cos(x) using a double angle formula?
The product of sin(x) and cos(x) can be simplified by working backwards from the sin(2x) formula. The identity is sin(2x) = 2sin(x)cos(x). To isolate the product sin(x)cos(x), you can simply divide both sides of the equation by 2. This gives a very useful simplification: sin(x)cos(x) = (1/2)sin(2x). This conversion is frequently used in calculus and equation solving to combine separate sin and cos terms into a single, more manageable trigonometric function.
