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Sin 2x Formula

Sin 2x 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 sin 2x formulas and can be verified by using basic trigonometric formulas.

sin 2x = 2 sin x cos x
sin 2x = 2 √(1 - cos²x) cos x
sin 2x = 2 sin x √(1 - sin²x)
sin 2x = (2tan x)/(1 + tan²x)

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 sin 2x and sin^2x in terms of various trigonometric functions using different trigonometric formulas and hence, derive the formulas.

1.	What is Sin 2x?
2.	Sin 2x Formula
3.	Derivation of Sin 2x Identity
4.	Sin 2x Formula in Terms of Tan
5.	Sin^2x (Sin Square x)
6.	Sin^2x Formula
7.	FAQs on Sin 2x Formula

What is Sin 2x?

Sin 2x 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. Sin 2x formula can be expressed in different forms using different formulas in trigonometry. The most commonly used formula of sin 2x is twice the product of sine function and cosine function which is mathematically given by, sin 2x = 2 sinx cosx. We can express sin 2x in terms of sine/cosine/tangent function alone as well.

The domain of sin 2x is the set of all real numbers. As the range of sine function is [-1, 1], the range of sin 2x is also [-1, 1].

Sin 2x Formula

The sin 2x 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 sin 2x:

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

These are the main formulas of sin 2x. 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 sin 2x in terms of cos and sin are:

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

sin 2 x Formula says 2 sin x cos x or 2 tan x over 1 plus tan square x

Derivation of Sin 2x Identity

To derive the formula for sin 2x, 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 sin 2x 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

⇒ sin 2x = 2 sin x cos x

Hence, we have derived the formula of sin 2x.

Sin 2x Formula in Terms of Tan

We can write the formula of sin 2x in terms of tan or tangent function only. For this, let us start with the sin 2x formula.

sin 2x = 2 sin x cos x

Multiply and divide the above equation by cos x. Then

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 one of the 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).

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 sin²x formula, we will use the trigonometric identities sin²x + cos²x = 1 and the double angle formula of cosine function given by cos 2x = 1 - 2 sin²x. Using these identities, we can express the formulas of sin²x in terms of cos x and cos 2x.

sin²x = 1 - cos²x
sin²x = (1 - cos 2x)/2

Let us derive the formulas stepwise below:

Sin^2x Formula in Terms of Cosx

We have the Pythagorean trigonometric identity given by sin²x + cos²x = 1. Using this formula and subtracting cos²x from both sides of this identity, we can write it as sin²x + cos²x -cos²x = 1 - cos²x which implies sin²x = 1 - cos²x. This formula of sin²x 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 cos 2x = 1 - 2sin²x. Using this formula and interchanging the terms, we can write it as 2 sin²x = 1 - cos2x ⇒ sin²x = (1 - cos2x)/2. Hence the formula of sine square x using the cos2x formula is sin²x = (1 - cos 2x)/2. This formula is used to solve complex integration problems.

Important Notes on Sin 2x:

Sin 2x formula is called the double angle formula of the sine function.
The important formulas of sin 2x is sin 2x = 2 sin x cos x and sin 2x = (2tan x)/(1 + tan²x)
The formula for sin²x is sin²x = 1 - cos²x and sin²x = (1 - cos 2x)/2

☛ Related Articles:

Sin 2x Examples

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

Solution:

We have 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

Answer: The value of sin⁡ 2A = 24/25.
Example 2: If sin A = 2/3 where A is in quadrant I, then find the value of sin 2A.

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 sin 2x formula,

sin 2A = 2 sin A cos A

= 2 (2/3) (√5/3)

= 4√5/9

Answer: The value of sin 2A = 4√5/9.
Example 3: If tan A = 4/3, find the value of sin 2A.

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)

sin 2A = (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 sin²x, we will use its formula sin²x = (1 - cos 2x)/2 to simplify the problem.

∫ sin²x dx = ∫[(1 - cos 2x)/2] dx

= (1/2) ∫dx - (1/2) ∫cos 2x dx

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

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

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Sin 2x Practice Questions

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FAQs on Sin 2x Formula

What are Sin 2x Identities?

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

What is Sin 2A 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.

What is the Period of Sin 2x?

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

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 Sin Square A Minus Sin Square B Formula?

Sin square a minus sin square b formula says sin²A - sin²B = sin (A + B) sin (A + B).

Is Sin 2x Equal to 2 Sin x?

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

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.

What is Sin Square x Formula?

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.

What is Sin 2x in Terms of Tan?

The general formula of sin 2x is sin 2x = 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 sin 2x in terms of tan.

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