Sin 2x cos 2x: Value, Derivative, and Integral Derivation

Content Curator

Sin 2x cos 2x is a trigonometric identity that is necessary for answering a variety of trigonometric questions.

An identity is a mathematical equation that always holds true.
A trigonometric identity is a true identity for all right-angled triangles that contain trigonometric functions.
The sine function of an angle represents the ratio between the opposite side and its hypotenuse.
The cosine function represents the ratio between the hypotenuse and its adjacent angels.

Table of Content

Value of Sin 2x Cos 2x
Derivation of Sin 2x Cos 2x
Derivative of Sin 2x Cos 2x
Integral of Sin 2x Cos 2x
Things to Remember
Sample Questions

Key Terms: Sine function, Cosine function, Trigonometric functions, Hypotenuse, Sin x, Cos x, Trigonometric identity, Derivative, Integral

Value of Sin 2x Cos 2x

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The value of sin 2x × Cos 2x is:

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)

Value of Sin 2x Cos 2x

Also check:

Related Concepts
Trigonometric Ratios	Cos2x Formula	Trigonometry Values
Introduction to Trigonometry	Sin Squared X Formula	Sin 120

Derivation of Sin 2x Cos 2x Value

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To find the value of sin2x × Cos 2x, the trigonometric double angle formulas are used. For the derivation, the values of sin 2x and cos 2x are used.

From trigonometric double-angle formulas,

Sin 2x = 2 sin x cos x ….(i)

And,

Cos 2x = Cos²x − Sin²x …,(ii)

But Sin²x + Cos²x = 1

⇒ Sin²x = 1 – Cos²x

On substituting in equation (ii), we get

Cos 2x = Cos²x − (1 – Cos²x)

⇒ Cos 2x = 2 cos²x − 1 ….(iii)

Also, Cos 2x = 1 − 2Sin²x ….(iv)

Multiplying equation (i) and (iii), we get

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

⇒ Sin 2x Cos 2x = 4 Sinx Cos³x − 2 Sin x Cos x

⇒ Sin 2x Cos 2x = 2 Cosx (2 Sinx Cos²x − Sinx)

Multiplying equation (i) and (iv), we get

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

⇒ Sin 2x Cos 2x = 4 Sinx Cosx − 4 Sin³ x Cos x

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

Therefore, we get

Sin 2x Cos 2x = 2Cos x (2Sin x Cos²x − Sin x) Or,

Sin 2x Cos 2x = 2Cos x (Sin x – 2Sin³x)

Derivative of Sin 2x Cos 2x

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The derivative of Sin 2x Cos 2x is given as

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

Proof

We can write

Sin (2x) cos (2x) = 1/2[2sin (2x) cos(2x)]

⇒ Sin (2x) cos (2x) = 1/2 Sin (4x)

Now, differentiate the both sides of the above equation with respect to x, we get

d/dx (Sin 2x Cos 2x) = d/dx [1/2 Sin(4x)]

⇒ d/dx (Sin 2x Cos 2x) = 1/2[d/dx{ Sin(4x) }]

⇒ d/dx (Sin 2x Cos 2x) = 1/2[ Cos (4x) d/dx (4x) ]

⇒ d/dx (Sin 2x Cos 2x) = 1/2[ Cos (4x) (4) ]

⇒ d/dx (Sin 2x Cos 2x) = 2 Cos (4x)

Integral of Sin 2x Cos 2x

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The integral of Sin 2x Cos 2x

∫ (Sin 2x Cos 2x) = (Sin 2x)²/ 4 + C

Proof

Let sin 2x = z

Therefore, dz/dx = 2Cos (2x)

⇒ dx = dz/[2Cos(2x)]

Now, we have

∫z Cos(2x) dx = ∫z • Cos(2x) • dz/2cos 2x

Here, Cos 2x can be canceled out.

Hence,

∫z Cos(2x)dx = ∫(z • du/2)

⇒ ∫z Cos(2x)dx = 1/2 [∫z dz]

⇒ ∫u Cos(2x)dx = 1/2 (z²/2) + c

⇒ ∫u Cos(2x)dx = z²/4 + C

⇒ ∫ (Sin 2x Cos 2x) = (Sin 2x)²/ 4 + C

Also check:

Related Concepts
Definite Integral Formula	Isosceles Triangle Theorems	First Order Differential Equation
Domain and Range of Trigonometric Functions	Law of Sines	Product to Sum Formulas With Solved Examples

Things to Remember

An identity is an equation that always holds true.
A true identity for all right-angled triangles containing trigonometric functions is a trigonometric identity.
The sin function of an angle is the ratio of the opposite side to the hypotenuse.
The cos function represents the ratio of the hypotenuse to its adjacent angels.
The value of Sin 2x Cos 2x is 2Cos x (Sin x – 2Sin³x).

Sample Questions

Ques. What is Sine Function? (1 Mark)

Ans. The sine function in trigonometry is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.

Ques. What is Cosine Function? (1 Mark)

Ans. The cosine function is defined in a right triangle as the ratio of the length of the adjacent side to the length of the longest side, i.e. the hypotenuse.

Ques. Is sine function even or odd? (1 Mark)

Ans. The sine function is an odd function whereas cosine is an even function.

Ques. Find the general solution for each of the following equations: sin 2x + cos x = 0. (3 Marks)

Ans. The given equation is,

sin 2x + cos x = 0

2sin x cos x + cos x = 0 [By double angle formulas, sin 2A = 2sin A cos A]

cos x(2sin x + 1) = 0

cos x = 0 or 2sin x + 1 = 0

cos x= 0 or sin x = -1/2

We know that cos x = 0 when x = π/2 and sin x = -1/2 when x = 7π/6. Thus,

cos x = cos π/2 or sin x = sin 7π/6, where n∈Z.

x = [(2n + 1)π/2] or x = [nπ + (-1)n7π/6], where n∈Z.

Ques. If sin θ + sin 2θ + sin 3θ = sin α, cos θ + cos 2θ + cos 3θ = cos α, then theta is equal to? (3 Marks)

Ans. Given,

sin θ + sin 2θ + sin 3θ = sin α

The given equation can be written as:

sin 2θ(2 cos θ + 1) = sin α….(1)

Also, given:

cos θ + cos 2θ + cos 3θ = cos α

Similarly, the above equation can be written as:

cos 2θ(2 cos θ + 1) = cos α….(2)

Now, divide (1) by (2), we get

[sin 2θ(2 cos θ + 1)]/ [cos 2θ(2 cos θ + 1)] = sin α/cos α

tan 2θ = tan α

2θ = α

θ = α/2

Ques. Find the value of (sin 8x + 7sin 6x + 18 sin 4x + 12 sin 2x)/ (sin 7x+6 sin 5x+12 sin 3x). (3 Marks)

Ans. Given expression: (sin 8x + 7sin 6x + 18 sin 4x + 12 sin 2x)/ (sin 7x+6 sin 5x+12 sin 3x).

The numerator of the given expression is sin 8x + 7sin 6x + 18 sin 4x + 12 sin 2x.

Now, simplify the expression:

= (sin 8x + sin6x) + 6(sin 6x + sin 4x) + 12(sin 4x + sin 2x)

= 2sin7x cos x + 12sin5x cos x + 24sin3x cos x

= 2cos x (sin7x + 6sin5x + 12sin3x)

Substituting the simplified numerator expression in the given expression, we get

(sin 8x + 7sin 6x + 18 sin 4x + 12 sin 2x)/ (sin 7x+6 sin 5x+12 sin 3x)=2cos x (sin7x + 6sin5x + 12sin3x)/(sin 7x+6 sin 5x+12 sin 3x)

= 2 cos x.

Hence, the value of (sin 8x + 7sin 6x + 18 sin 4x + 12 sin 2x)/ (sin 7x+6 sin 5x+12 sin 3x) is 2 cos x.

Ques. Derive the derivative of sin 2x cos 2x. (2 Marks)

Ans. Sin 2x cos 2x = 1/2 (2 sin 2x cos 2x) (Or) 1/2 sin 4x

By differentiating the given function:

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

Ques. Derive the integral of sin 2x cos 2x. (2 Marks)

Ans. 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

∫ydy

= ½ y²/2 + c

= y²/4 + C

Ques. Find the general solution of the equation sin 2x + cos x =0 (3 Marks)

Ans. Sin 2x + cos x =0

Putting sin 2x + cos x =0

2 sin x cos x + cos x =0

Cosx (2sinx= -1) =0

Hence, 2 sin x +1=0

2sin x =-1

Sin x =-½

Ques. Find the general solution for each of the following equations: sin 2x + cos x = 0 (1 Mark)

Ans. The general solutions of sin 2x + cos x = 0 are x = [(2n + 1)π/2] or [nπ + (-1)n7π/6], where n∈Z.

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Sin 2x cos 2x: Value, Derivative, and Integral Derivation

Value of Sin 2x Cos 2x

Derivation of Sin 2x Cos 2x Value

Derivative of Sin 2x Cos 2x

Proof

Integral of Sin 2x Cos 2x

Proof

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

3.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

4.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

5.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

6.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

Comments

SUBSCRIBE TO OUR NEWS LETTER

Sin 2x cos 2x: Value, Derivative, and Integral Derivation

Value of Sin 2x Cos 2x

Derivation of Sin 2x Cos 2x Value

Derivative of Sin 2x Cos 2x

Proof

Integral of Sin 2x Cos 2x

Proof

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

3.Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

4. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

5.At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.On the basis of the above information, answer the following questions :

6. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

3.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

4.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

5.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

6.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).