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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.
|1.||What is Sin2x?|
|3.||Derivation of Sin 2x Identity|
|4.||Sin2x Formula in Terms of Tan|
|5.||Sin^2x (Sin Square x)|
|7.||FAQs on Sin2x Formula|
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.
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
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.
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.
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Example 1: If cos A = 3/5 where A is in quadrant I, then find the value of sin2A.
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 sin2x formula, sin2x = 2 sin x cos x. From this,
sin2A = 2 sin A cos A
= 2 (4/5) (3/5)
Answer: The value of sin2A = 24/25.
Example 2: If sin A = 2/3 where A is in quadrant I, then find the value of sin2A.
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)
Answer: The value of sin2A = 4√5/9.
Example 3: If tan A = 4/3, find the value of sin2A.
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.