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
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].
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)
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.
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).
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.
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:
- Derivative of Sin 2x
- Integral of Sin 2x and Sin^2x
- Derivative of Cos2x
