Sin2x Formula: Derivation, Examples & Uses

Sin2x Formula is one of the most important Identiies in Trigonometry. Double angle formulae are trigonometric formulae such as Sin2x, Cos 2x, and Tan 2x.The Sin 2x Formula is one of the double-angle formulas used to calculate the sine of an angle with a double value.  The sin2x formula can be represented in different forms using various trigonometric functions. The most common sin2x formula is sin2x = 2 sinx cosx. In this article, we are going to know the concept of the Sin2x formula, along with its derivation, and examples.

What is sin2x?

Sin 2x is a trigonometric identity that determines a variety of trigonometric expressions with double angles. We are all familiar with the term sin, which is defined as the ratio of the length of the opposing side (of the angle) to that of the length of the hypotenuse in a right-angled triangle. Sin2x is the product of the sine and cosine functions, denoted as sin2x = 2sin x cosx. The domain of sin 2x is the set of all real numbers. As the range of the sine function is [-1, 1], the range of sin 2x is also [-1, 1].

What is Sin2x Formula?

Sin 2x formula is used to solve various trigonometric, integration, and differentiation problems in trigonometry. The Sin 2x formula can be represented in a variety of ways using various basic trigonometric identities. The most frequent Sin2x formula is twice the product of the sine function and the cosine function, which is expressed mathematically as

sin2x = 2 sinx cosx.

Sin2x formula

There are several types of sin 2x formulae, all of which may be derived using simple trigonometric formulas.The Sin2x formula has two basic and popular expressions:

• sin 2x = 2 sin x cos x
• sin 2x = 2 √(1 – cos²x) cos x

Sin2x Formula can be written as sin x (or cos x) alone using the trigonometric formula Sin²x + cos²x = 1. As a result, the Sin2x formula is as follows in terms of cos and sin:

• sin 2x = 2 sin x √(1 – sin²x)
• sin 2x = (2tan x)​/(1 + tan²x)

Sin2x Formula Derivation

The Derivation of Sin2x Formulas helps us to remember easily. Using the angle sum formula of sin, we can easily find out the Sin 2x formula. Let us look at the sin2x Formula derivation step by step:

We know that Sin(A + B) = sin A cos B + sin B cos A .

Now, in the above formula substitute A = B = x.

Putting A = B = x. we get,

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

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

As a result, using this method we can derive the sin2x formula.

Sin2x Formula in Terms of Tan

The sin2x formula can alternatively be expressed using the tan function. Let’s look at how Sin2x formula is expressed in terms of tan x.

sin 2x = 2 sin x cos x

Multiplying and dividing by cos x.

Then

sin 2x = (2 sin x cos²x)/(cos x)

= 2 (sin x/cosx ) × (cos²x)

Using sin x/cos x = tan x , cos x = 1/(sec x) formulas we get,

sin 2x = 2 tan x × (1/sec²x)

sin 2x = (2tan x)​/(1 + tan²x)  {using, sec²x = 1 + tan²x}

Hence, the sin2x formula in terms of tan is

sin 2x = (2tan x)​/(1 + tan²x)

Sin2x Formula in Terms of Cos

The sin2x formula can alternatively be expressed using the Cos function. Check out the derivation of the Sin2x Formula in terms of cos x.

we Know that, sin²x + cos²x = 1.

Or, sinx = √(1 – cos²x)

We know that The sin 2x Formula which is sin 2x= 2 sin x cos x

In the equation , we can use sin x = √(1 – cos²x)

So, we get the Sin2x formula in terms of Cos x is as follows

sin 2x = 2 √(1 – cos²x) × cos x

Sin2x Formula in Terms of Sin

The sin2x formula can also be expressed using the sin function. Check out the derivation of the Sin2x Formula in terms of sin x.

we Know that, sin²x + cos²x = 1.

Or, cosx = √(1 – sin²x)

We know that The sin 2x Formula which is sin 2x= 2 sin x cos x

In the equation, we can use cosx = √(1 – sin²x)

So, we get the Sin2x formula in terms of Sin x is as follows

sin 2x = (2 sin x )× √(1 – sin²x)

Sin2x Formula Based Solved Examples

Example 1: If sec x = 5/3, determine the value of Sin2x.

Solution; Accordingly the question, sec x = 5/3.
Cos x = 3/5 and sin x = 4/5 according to Pythagoras Theorem.
We obtain using the Sin2x Formula
Sin2x = 2 sin x cos x
= 2 (4/5) (3/5)
= 24/25
Therefore,  the value of Sin2x if sec x = 5/3 is 24/25.

Example 2: Find the sin 90-degree value using the Sin2x Formula.
Solution: The Sin2x Formula must be applied to determine the value of sin 90 degrees.First we need to find the value of x.
2x = 90°
x = 90°/2
x = 45°

Now, Substituting the value of x into the Sin2x Formula,
As per the Sin2x formula we get,
sin 2x = 2 sin x cos x
Sin (2 x 45°) = 2sin45° cos45°
We already know that sin45° = 1/√2 and cos 45° = 1/√2. putting these values we obtain,
Sin 90°=2×1/√2 x 1/√2
Sin 90° = 1

Therefore, the value of sin 90° is 1.

Example 3: Use the formula to 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.
As per the Sin2x formula we get,
sin 2x = 2 sin x cos x

Putting he given Sinx value and Cos x value,
sin2x = 2 (4/5) (3/5)
Sin 2x = 24/25

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