Sin2x Formula

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Sin2x formula is one of the double angle formulas in trigonometry. Using this formula, we can find the sine of the angle whose value is doubled. We are familiar that sin is one of the primary trigonometric ratios that is defined as the ratio of the length of the opposite side (of the angle) to that of the length of the hypotenuse in a right-angled triangle. There are various formulas related to sin2x and can be verified by using basic trigonometric formulas. As the range of sin function is [-1, 1], the range of sin2x is also [-1, 1].

Further in this article, we will also explore the concept of sin^2x (sin square x) and its formula. We will express the formulas of sin2x and sin^2x in terms of various trigonometric functions using different trigonometric formulas and hence, derive the formulas.

What is Sin2x?

Sin2x is a trigonometric formula in trigonometry that is used to solve various trigonometric, integration, and differentiation problems. It is used to simplify the various trigonometric expressions. Sin2x formula can be expressed in different forms using different formulas in trigonometry. The most commonly used formula of sin2x is twice the product of sine function and cosine function which is mathematically given by, sin2x = 2 sinx cosx. We can express sin2x in terms of tangent function as well.

Sin2x Formula

The sin2x formula is the double angle identity used for sine function in trigonometry. Trigonometry is a branch of mathematics where we study the relationship between the angles and sides of a right-angled triangle. There are two basic formulas for sin2x:

  • sin2x = 2 sin x cos x (in terms of sin and cos)
  • sin2x = (2tan x)​/(1 + tan2x) (in terms of tan)

These are the main formulas of sin2x. But we can write this formula in terms of sin x (or) cos x alone using the trigonometric identity sin2x + cos2x = 1. Using this trigonometric identity, we can write sinx = √(1 – cos2x) and cosx = √(1 – sin2x). Hence the formulas of sin2x in terms of cos and sin are:

  • sin2x = 2 √(1 – cos2x) cos x (sin2x formula in terms of cos)
  • sin2x = 2 sin x √(1 – sin2x) (sin2x formula in terms of sin)

Derivation of Sin 2x Identity

To derive the formula for sin2x, the angle sum formula of sin can be used. The sum formula of sin is sin(A + B) = sin A cos B + sin B cos A. Let us see the derivation of sin2x step by step:

Substitute A = B = x in the formula sin(A + B) = sin A cos B + sin B cos A,

sin(x + x) = sin x cos x + sin x cos x

⇒ sin2x = 2 sin x cos x

Hence, we have derived the formula of sin2x.

Sin2x Formula in Terms of Tan

We can write the formula of sin2x in terms of tan or tangent function only. For this, let us start with the sin2x formula.

sin2x = 2 sin x cos x

Multiply and divide the above equation by cos x. Then

sin2x = (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

sin2x = 2 tan x × (1/sec2x)

Using one of the Pythagorean trigonometric identities, sec2x = 1 + tan2x. Substituting this, we have

sin2x = (2tan x)​/(1 + tan2x)

Therefore, the sin2x formula in terms of tan is sin2x = (2tan x)​/(1 + tan2x).

Sin^2x (Sin Square x)

In this section of the article, we will discuss the concept of sin square x. We have two formulas for sin^2x which can be derived using the Pythagorean identities and the double angle formulas of the cosine function. Sin^2x formulas are used to solve complex integration problems and to prove different trigonometric identities. In the next section, we will derive and explore the formulas of sin square x.

Sin^2x Formula

To derive the sin^2x formula, we will use the trigonometric identities sin^2x + cos^2x = 1 and double angle formula of cosine function given by cos2x = 1 – 2 sin^2x. Using these identities, we can express the formulas of sin^2x in terms of cosx and cos2x. Let us derive the formulas stepwise below:

Sin^2x Formula in Terms of Cosx

We have the Pythagorean trigonometric identity given by sin^2x + cos^2x = 1. Using this formula and subtracting cos^2x from both sides of this identity, we can write it as sin^2x + cos^2x -cos^2x = 1 – cos^2x which implies sin^2x = 1 – cos^2x. Hence, the formula of sin square x using Pythagorean identity is sin^2x = 1 – cos^2x. This formula of sin^2x is used to simplify trigonometric expressions.

Sin^2x Formula in Terms of Cos2x

Now, we have another trigonometric formula which is the double angle formula of the cosine function given by cos2x = 1 – 2sin^2x. Using this formula and interchanging the terms, we can write it as 2 sin^2x = 1 – cos2x ⇒ sin^2x = (1 – cos2x)/2. Hence the formula of sine square x using the cos2x formula is sin^2x = (1 – cos2x)/2. This formula of sin^2x is used to solve complex integration problems. Therefore, the two basic formulas of sin^2x are:

  • sin^2x = 1 – cos^2x ⇒ sin2x = 1 – cos2x
  • sin^2x = (1 – cos2x)/2 ⇒ sin2x = (1 – cos2x)/2

Important Notes on Sin2x

  • The important formula of sin2x is sin2x = 2 sin x cos x and sin2x = (2tan x)​/(1 + tan2x)
  • The formula for sin^2x is sin^2x = 1 – cos^2x and sin^2x = (1 – cos2x)/2
  • Sin2x formula is called the double angle formula of the sine function.
 

Sin2x Examples

Example 2: If sin A = 2/3 where A is in quadrant I, then find the value of sin2A.

Solution:

We have

sin2A + cos2A = 1

cos2A = 1 − sin2A

cos A = ±√(1 − sin2A)

cos A = ±√(1 − (2/3)2)

cos A = ±√(5/9)

cos A = ±√5/3

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

cos A = √5/3

Using the sin2x formula,

sin2A = 2 sin A cos A

= 2 (2/3) (√5/3)

= 4√5/9

Answer: The value of sin2A = 4√5/9.

Example 3: If tan A = 4/3, find the value of sin2A.

Solution:

We have

sin 2x = 2tan(x)​/(1 + tan2(x))

So sin 2A = 2tan(A)​/(1 + tan2(A))

Putting the value of tan A = 4/3, we get

Sin 2A = 2(4/3)​/(1 + (4/3)2)

Sin2A = (8/3)/(25/9)

Sin 2A = 24/25

Answer: sin 2A = 24/25.

Example 4: Evaluate the integral of sin square x.

Solution: To find the integral of sin^2x, we will use its formula sin^2x = (1 – cos2x)/2 to simplify the problem.

∫ sin^2x dx = ∫[(1-cos2x)/2] dx

= (1/2) ∫dx – (1/2) ∫cos2x dx

= x/2 – (1/4) sin2x + C [Because the integral of cos2x is (1/2) sin2x]

Answer: The integral of sin^2x is x/2 – (1/4) sin2x + C.

FAQs on Sin2x Formula

What is Sin2x Formula?

Sin2x formula is the double angle formula of sine function and sin 2x = 2 sin x cos x is the most frequently used formula. But sin2x in terms of tan is sin 2x = 2tan(x)​/(1 + tan2(x)).

What is the Period of Sin2x?

The period of sin bx is (2π)/b in general. So the period of sin2x is (2π)/2 = π which implies the value of sin2x repeats after every π radians.

What is Sin2A in Terms of Cos?

The general formula of sin2A is, sin2A = 2 sin A cos A. Using sin2A + cos2A = 1, we get sin A = √(1 – cos2A). Substituting this in the given formula, sin2A = 2 √(1 – cos2A) cos A. This formula is in the terms of cos or cosine function only.

How to Prove Sin 2x Formula?

From the sum formula of sin, we have sin (A + B) = sin A cos B + cos A sin B. Substituting A = B = x here, we get, sin 2x = 2 sin x cos x.

What is Sin2A in Terms of Sin?

The general formula of sin 2A is, sin 2A = 2 sin A cos A. Using sin2A + cos2A = 1, we get cos A = √(1 – sin2A). Substituting this in the above formula, sin2A = 2 sin A √(1 – sin2A). This formula is in the terms of sin or sine function only.

Is Sin2x Equal to 2 Sin x?

No, sin2x is not equal to 2 sin x. In fact, sin2x = 2 sin x cos x from the double angle formula of sin.

What is Sin2x in Terms of Tan?

The general formula of sin2x is sin2x = 2 sin x cos x = 2 (sin x cos2x)/(cos x) = 2 (sin x/cos x) (1/sec2x) = (2 tan x)/(1 + tan2x). This is sin2x in terms of tan.

What is the Formula of Sin^2x?

Sin^2x Formula can be derived from trigonometric identities sin^2x + cos^2x = 1 and cos2x = 1 – 2sin^2x. Using these formulas, the formula for sin^2x are sin^2x = 1 – cos^2x and sin^2x = (1 – cos2x)/2.

Sin 2x Derivation Formula

The formula for sin 2x can be derived by using the sum angle formula for the sine function.

Using Trigonometric Identities, sin (x + y) = sin x cos y + cos x sin y

In order to find sine for double angle, we must put x = y

Putting x = y we get,

sin (x + x) = sin x cos x + cos x sin x

sin 2x = sin x cos x + sin x cos x

sin 2x = 2 sin x cos x

This derives the formula for the double angle of the sine ratio.

Sin2x Formula in Terms of Cos

sin 2x can also be given in terms of cos function. Let’s take a look at how Sin 2x is given in terms of cos x

sin 2x = 2 sin x cos x …(1)

we know that sin x = √(1 – cos2x) using this in eq (1)

sin 2x = 2 √(1 – cos2x) × cos x 

This is the required formula for Sin 2x in terms of Cos x.

Sin2x Formula in Terms of Sin

sin 2x can also be given in terms of sin function. Let’s take a look at how Sin 2x is given in terms of sin x

sin 2x = 2 sin x cos x …(1)

we know that cos x = √(1 – sin2x) using this in eq (1)

sin 2x = (2 sin x )× √(1 – sin2x) 

This is the required formula for Sin 2x in terms of Sin x.

Sin2x

Sin2x formulas are used to solve complex mathematical problems, they are also used to simplify trigonometric identities. Two formulas for sin2x can be derived using the Pythagorean Theorem and the double angle formulas of the cosine function.

Sin2x Formula

For the derivation of the sin2x formula, we use the trigonometric identities sin2x + cos2x = 1 and the double angle formula of cosine function cos 2x = 1 – 2 sin2x. Using these identities, sin2x can be expressed in terms of cos2 x and cos2x. Let us derive the formulas:

Sin2x Formula in Terms of Cos x

We know that, using trigonometric identities,

sin2x + cos2x = 1 using the equation and sending cos2x to the left-hand side which changes its sign we get,

sin2x = 1 – cos2x

Sin2x Formula in Terms of Cos 2x

We know that, using the double-angle formula,

cos 2x = 1 – 2sin2x using the equation and separating sin2x to one side we get,

sin2x = (1 – cos 2x) / 2

Therefore, the two basic formulas of sin2x are:

sin2x = 1 – cos2

sin2x = (1 – cos 2x) / 2

Important Formulas

sin 2x = 2 sin x cos x
sin 2x = (2tan x)​/(1 + tan2x)

Other Formulas

sin2x  = 1 – cos2
sin2x = (1 – cos2x)/2

Solved Examples on Sin 2x Formula

Example 1. If sin x = 3/5, find the value of sin 2x using the formula.

Solution:

We have, sin x = 3/5.

Clearly, cos x = 4/5.

Using the formula we get,

sin 2x = 2 sin x cos x

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

= 24/25

Example 2. If cos x = 12/13, find the value of sin 2x using the formula.

Solution:

We have, cos x = 12/13.

Clearly, sin x = 5/13.

Using the formula we get,

sin 2x = 2 sin x cos x

= 2 (5/13) (12/13)

= 120/169

Example 3. If tan x = 12/5, find the value of sin 2x using the formula.

Solution:

We have, tan x = 12/5.

Using the formula we get,

sin2x = (2tan x)​/(1 + tan2x).

         = 2 × (12/5) / {1 + (12/5)2}

         = 120/169

Example 4. If cosec x = 17/8, find the value of sin 2x using the formula.

Solution:

We have, cosec x = 17/8.

Clearly sin x = 8/17 and cos x = 15/17.

Using the formula we get,

sin 2x = 2 sin x cos x

= 2 (8/17) (15/17)

= 240/289

Example 5. If cot x = 15/8, find the value of sin 2x using the formula.

Solution:

We have, cot x = 15/8

tan x = 1 / cot x = 1 / (15/8) 

        = 8 / 15

Using the formula we get,

sin2x = (2tan x)​/(1 + tan2x).

         = 2 × (18 / 15) / {1 + (18 / 15)2}

        = 240/289

Example 6. If cosec x = 13/12, find the value of sin 2x using the formula.

Solution:

We have, cosec x = 13/12.

Clearly sin x = 12/13 and cos x = 5/13 (using pythagoras theorem)

Using the formula we get,

sin 2x = 2 sin x cos x

          = 2 (12/13) (5/13)

         = 120/169

Example 7. If sec x = 5/3, find the value of sin 2x using the formula.

Solution:

We have, sec x = 5/3.

Clearly cos x = 3/5 and sin x = 4/5 (using pythagoras theorem)

Using the formula we get,

sin 2x = 2 sin x cos x

          = 2 (4/5) (3/5)

          = 24/25

FAQs on Sin 2x Formula

Question 1: What is the differentiation of Sin 2x?

Answer:

The differentiation of sin 2x is 2cos 2x

Question 2: What is the integration of Sin2x?

Answer: 

The integration of sin 2x is (-cos 2x) / 2

Question 3: What is the Sin 2x formula in terms of the tan function?

Answer:

Sin 2x formula in terms of the tan function is sin2x = (2tan x)​/(1 + tan2x).

Question 4: Write the formula for tan 2x.

Answer:

The formulas used for tan 2x are:

  • tan2x = 2tan x / (1−tan2x)
  • tan2x = sin 2x/cos 2x

Question 5: Write the formula for cos 2x.

Answer:

The formulas used for cos 2x are:

  • cos2x = cos2x – sin2x
  • cos2x = 2cos2x – 1
  • cos2x = 1 – 2sin2x
  • cos2x = (1 – tan2x)/(1 + tan2x)

Derivation of Sin2x Formula

Before going into the actual proof, first, let us take a look at the formula itself.

Sin 2x = 2 sinx cosx

Observe that the sin2x formula is a product of sinx and cosx. We will start by using the known formula in which sin and cos are multiples of each other. This approach leads to a formula that we know as the angle sum formula of sin.

Sin(a+b) = Sin a Cos b + Cos a Sin b

where a and b are angles.

We can replace a and b both as x which gives us,

Sin(x+x) = Sin x Cos x + Cos x Sin x

which can be written as,

Sin(2x)= 2Sin x Cos x

Hence Proved.

Use of Sin2x All Formula

These double angle formulas and to be more precise cos 2x and sin 2x formulas are used in the large problems of integration and differentiation. Apart from pure mathematics, they are also used in real-life problems of height and distance. The simplification of big problems will make it easier for us to solve them. This simplification is done by double angle formulas of cos 2x and sin 2x formula.

Examples Based on sin2x Formula

Question: Find the Value 2sinx sin2x Formula in Terms of Cos.

Answer: We can simplify the given expression by substituting the value of sin 2x. We know that 

Sin (2x) = 2Sin x Cos x

On substituting the value we get,

2sin x sin2x =2sinx 2sinxcosx

2sinxsin2x =4sin2xcosx

Since we need to get this expression in cos we can use the identity Sin2θ + Cos2θ = 1. We get,

2sinxsin2x = 4(1-cos2x)cosx

2sinxsin2x = 4cosx-4cos3x

Question: Find the Value of sin90o. Use the Double Angle Formula For that.

Answer: We know the double angle formula of sin which is the formula of sin2x as 2sinxcosx

In order to find the value of sin90o, we have to use this formula. We can do so by finding the correct value of x.

2x=90o

sin90o = 1

Hence the required value of sin90o is 1.

Question: Calculate the value of the given expressions sin75sin15.

Answer: The given expression,

Sin750 sin15o

= Sin ( 90o−15o) sin15o

= cos15o sin15o ascosx=sin(90∘−x)

Multiplying and dividing by 2

FAQs on Sin 2x Formula

1. What is the 2sinx sin2x Formula?

Here, we have to find the value of 2 sin x sin 2x. We know that sin2x=2sinxcosx.

Using the above formula in the given expression we will get,

2sinx . 2sinxcosx = 4sin2xcosx

And as sin2x = 1 – cos2x, the expression further becomes

⇒ 4 (1 – cos2 x) cos x = 4( cosx – cos3 x)

= 4 cosx – 4 cos3 x .

Thus the expression becomes as

2 sinx sin2x = 4 cos x – 4 cos3x . Hence this is the required result.

2. What is the cos 2x Formula? Prove it.

As proving the sin 2x formula, we can easily prove the cos 2x formula. As know the angle sum formula of cos is 

cos(a+b)=cos(a) cos(b) – sin(a) sin(b)

Where a and b are angled. 

Assuming that both a and b are equal and are equal the x.

Now replace a and b from x. We’ll get

cos(x+x)=cosx cosx – sinx sinx

Which gives us

cos2x=cos2x-sin2x

This is the required cos2x formula.

3. Prove that the value of sin90 is equal to 1 with the help of the sin 2x formula.

We know that the value of sin90o is 1 unit. i.e. sin90o  = 1. 

And in order to prove this statement true, let us first assume the 2x = 90o and from here, We got x = 450

Substituting the value in the sin 2x formula,  we will be able to prove the given statement.

sin 2x = 2 × sinx × cosx.

sin(2 x 45o) = 2 × sin45o × cos45o

We know that the value of these two terms are as sin45o  = 12 and cos45o = 12, substituting these values in the equation above.

sin90o  = 2 ×12 ×12

sin90o  = 2 ×12

sin90o = 1, hence proven the value of sin90o is equal to 1 (with the help of sin 2x formula)

4. Define sin 2x formula in terms of tan?

The formula of sin 2x can be written in terms of only tan (tangent function). Let us derive the sin 2x formula with only the tangent formula.

The original formula  sin 2x = 2 sinx cosx

Now, dividing and multiplying the equation with cos x

5. Provide an easy derivation of the sin 2x formula?

The sin value for the double angle is in the double the value of a product of sin and cos values of a single angle, i.e. sin 2x = 2 sinx cosx. This formula can easily be derived by using the addition formula for sin angles.

The additional formula for sin is given as:

sin(x+y) = sin(x) × cos(y) + cos(x) × sin(y), 

where x and y are typically two different angles, but if we assume the x = y, then the equation becomes as follows

sin(x+x) = sin(x) cos(x)  +  cos(x) sin(x) 

sin 2x = 2 sin x cos x.

Introduction to Sin 2Theta formula

Here we look at trigonometric formulae known as the double angle formulae. They are called so because it involves double angles trigonometric functions, i.e. sin 2x.

Deriving Double Angle Formulae for Sin 2

We start by recalling the addition formula to learn Sine double angle formula

sin(A + B) = sin A cos B + cos A sin B

Double Angle formula to get 2sinxcosx

Let’s see what happens if we let B equal to A.

After doing so, the first of these formulae becomes: sin(x + x) = sin x cos x + cos x sin x

so that sin2x = 2 sin x cos x.

And this is how our first double-angle formula, so called because we are doubling the angle (as in 2A).

Practice Example for Sin 2x

If we want to solve the following equation:

Sin 2x = sinx, -Π ≤ Π

We will follow the following steps:

Step 1) Use the Double angle formula

Sin 2x = 2 Sin x Cos x

Step 2) Let’s rearrange it and factorize

2Sinx Cosx – sinx = 0

Sin x(2 cos x -1) = 0

So, a) Sinx =0

or

b) cos2x -1 = 0

Step 3) Let’s consider Sin x = 0. (Refer to the graph)

We have two solutions as x = Π & x = 0.

Step 4) x = Π will be excluded as it isn’t in the interval mentioned in the question.

Step 5) from equation b, 2cosx -1 = 0

2 cos x = 1

Cos x =½

Cos x = 600 or Π/3

Step 6) Looking at the graph, we infer that

x = – Π/3 & x = Π/3

Example 1: Find the value 2sinx sin2x formula in terms of cos.

Ans: On substituting the value, we get,

2sinx sin2x = 2sinx 2sinx cosx

2sinx sin2x = 4sin2xcosx

Since we need to get this expression in Cos, we can use the identity sin2θ + cos2θ

 = 1.

We get,

2sinx sin2x = 4(cos2x) Cos x

2sinx sin2x = 4cosx-4cos3x

Example 2: Find the value of sin90 degree. Use the double angle formula.

Ans: We know the double angle formula of sin which is the formula of sin2x as 2sinxcosx

To find the value of sin90 degree, we have to use this formula. By finding the correct value of x, the value of sin90 degree can be found.

2x = 90 degree

X = 90°/2

X=45°

Now we have got the value of x. Let’s substitute the value into the sin2x formula

Sin (2x 45°) = 2sin45° cos45°

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

⇒Sin90°=2×1/√2 x 1/√2

sin90° = 1

Hence the required value of sin90° is 1

Question 1: Find the Value 2sinx sin2x Formula in Terms of Cos.

Answer: We can simplify the given expression by substituting the value of sin 2x. We know that  

Sin (2x) = 2Sin x Cos x 

On substituting the value we get,

2sin x sin2x =2sinx 2sinxcosx

2sinxsin2x =4sin2xcosx

Since we need to get this expression in cos we can use the identity Sin2θ + Cos2θ = 1. We get,

2sinxsin2x = 4(1-cos2x)cosx

2sinxsin2x = 4cosx-4cos3x

Question 2: Find value of sin90∘ using its double angle Sin2x  formula.

Answer:  Sin2x formula for double angle is 2Sinx Cosx

Putting  x=45∘

Sin(2×45∘)=2 Sin45∘ Cos45∘

Since  Sin45∘=1/√2

Cos45∘=1/√2

Therefore, by substituting the results will be:

Sin90∘=2×1/√2 ×1/√2

=2×1/2=1

Thus value of is Sin90∘1.

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