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Sin Double Angle Formula
Sin double angle formula in trigonometry is a sine function formula for the double angle 2θ. The formula for sin 2θ is used to simplify various problems in trigonometry. The sin double angle formula is one of the important double angle formulas in trigonometry. We can express sin of double angle formula in terms of different trigonometric functions including sin and cos, and tangent function. We know that sine of an angle is defined as the ratio of perpendicular and hypotenuse of a rightangled triangle.
In this article, we will discuss the concept of the sin double angle formula, prove its formula using trigonometric properties and identities, and understand its application. We shall solve a few examples using the different forms of the sin double angle formula for a better understanding of the concept.
1.  What is Sin Double Angle Formula? 
2.  Sin Double Angle Formula Proof 
3.  Sin Double Angle Formula in Terms of Tan 
4.  Sin Squared Double Angle Formula 
5.  FAQs on Sin Double Angle Formula 
What is Sin Double Angle Formula?
Sin Double Angle Formula is a trigonometric formula that is used to simplify various expressions and problems in trigonometry. It gives the value of the sine function for the double angle 2θ, that is, sin2θ. We can express the sin double angle formula in different forms and in terms of different trigonometric functions. It is one of the primary doubleangle formulas of trigonometry. Sin double angle formula can be expressed as twice the product of cosine and sine of the angle. Sin2θ formula can be expressed as:
 sin2θ = 2 sinθ cosθ
 sin2θ = 2tanθ / (1 + tan^{2}θ)
Sin Double Angle Formula Proof
Now that we know the two sin double angle formula, let us derive these formulas using trigonometric formulas and identities. To derive the first formula of sin2θ, we will use the sin A plus B formula given by, sin (a + b) = sin a cos b + cos a sin b. In this formula, substitute a = θ and b = θ. So, we have
sin (a + b) = sin a cos b + cos a sin b
⇒ sin (θ + θ) = sin θ cos θ + cos θ sin θ
⇒ sin2θ = 2 sinθ cosθ
Hence, we have proved the first sin double angle formula.
Sin Double Angle Formula in Terms of Tan
We will use the above sin double angle formula to express in terms of tan. We will prove the sin2θ formula in terms of the tangent function. Multiply and divide the formula sin2θ = 2 sinθ cosθ by cos θ. Then, we have
sin2θ = (2 sin θ cos^{2}θ)/(cos θ)
= 2 (sinθ / cosθ ) × (cos^{2}θ)
We know that sin θ/cos θ = tan θ and cos θ = 1/(sec θ). So
sin2θ = 2 tan θ × (1/sec^{2}θ)
Using one of the Pythagorean trigonometric identities, we have sec^{2}θ = 1 + tan^{2}θ. Substituting this, we have
sin2θ = (2tan θ)/(1 + tan^{2}θ)
Therefore, the sin double angle formula in terms of tan is sin2θ = (2tan θ)/(1 + tan^{2}θ).
Sin Squared Double Angle Formula
Sin squared double angle formula gives the trigonometric formulas for the expressions sin^{2}(2x). To express the sin^{2}(2x) formula, we just replace θ with 2x in the sin^{2}θ formula. So, first, let us write sin^{2}θ formula
 sin^{2}θ = 1  cos^{2}θ
 sin^{2}θ = (1/2) (1  cos2θ)
Now, simply replacing θ with 2x in the above formulas, we can have the sin squared double angle formulas as given below:
 sin^{2}2x = 1  cos^{2}2x
 sin^{2}2x = (1/2) (1  cos4x)
Important Notes on Sin Double Angle Formula
 Sin Double Angle Formula can be expressed in terms of different trigonometric functions.
 The formula for sin2θ can be expressed as:
 sin2θ = 2 sinθ cosθ
 sin2θ = 2tanθ / (1 + tan^{2}θ)
 We can prove the sin double angle formulas using the sin (A + B) formula and other trigonometric identities.
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Sin Double Angle Formula Examples

Example 1: Find the value of sin2x if cos x = 1/√2 and sin x = 1/√2 using the sin double angle formula.
Solution: We know that sin2θ = 2 sinθ cosθ, so we have
sin2x = 2 sinx cosx
= 2 × 1/√2 × 1/√2
= 2 × 1/2
= 1
Answer: sin2x = 1

Example 2: Calculate the sin 120° using the sin double angle formula in terms of tan.
Solution: Using sin2x formula in terms of tan, the value of sin120° is given by,
sin120° = 2 tan60° / (1 + tan^{2}60°)
= (2 × √3) / (1 + (√3)^{2})
= 2√3 / (1 + 3)
= 2√3 / 4
= √3 / 2
Answer: sin120° = √3/2

Example 3: Evaluate the value of cos x if sin x = 1/2 and sin2x = √3/2
Solution: To find the value of cos x if sin x = 1/2 and sin2x = √3/2, we will use the sin double angle formula. We have
sin2x = 2 sinx cosx
⇒ √3/2 = 2 × 1/2 × cos x
⇒ cos x = √3/2 × 1/2 × 2
= √3/2
Answer: cos x = √3/2
FAQs on Sin Double Angle Formula
What is Sin Double Angle Formula in Trigonometry?
Sin Double angle formula is an important formula in trigonometry which gives the value of the sine function for the double angle. It is an important double angle formula and its formula can be written in two ways:
 sin2θ = 2 sinθ cosθ
 sin2θ = 2tanθ / (1 + tan^{2}θ)
How to Use Sin Double Angle Formula?
We can use the sin double angle formula to find the value of the sine function for the double angle 2θ. Simply substitute the known values into the sin double angle formula to find the value of the unknown variable.
How to Derive Sin Double Angle Formula?
We can derive the sin double angle formula using the sum formula of sine and other trigonometric formulas and identities.
What is Sin Squared Double Angle Formula?
The sin squared double angle formulas as given below:
 sin^{2}2x = 1  cos^{2}2x
 sin^{2}2x = (1/2) (1  cos4x)
When to Use Sin Double Angle Formula?
We can use the sin double angle formula to find the value of sin2x when the value of the trigonometric functions is known for the angle x.
What is Sin Double Angle Formula in Terms of Tan?
We can express the sin double angle formula in terms of tan as sin2θ = 2tanθ / (1 + tan^{2}θ).
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