2 sin a cos a is a trigonometric formula that is equal khổng lồ the sine of angle 2a, i.e., it is given by 2 sin a cos a = sin 2a. It is one of the important trigonometric identities that is used to solve various trigonometric & integral problems. 2 sin a cos a formula is also called the double angle formula of sine function as it is equal lớn sin 2a, where 2a is twice the angle a. This formula can also be expressed in terms of chảy a.

Bạn đang xem: Proofs of trigonometric identities i, sin 2x = 2sin x cos x

Let us explore the 2 sin a cos a formula, derive the formula using the sin (a + b) formula, & understand its application khổng lồ solve different mathematical problems. We will also some a few examples using 2 sin a cos a formula for a better understanding of its application.

 1 What is 2 Sin a Cos a Formula? 2 Derivation of 2 Sin a Cos a Formula 3 2 Sin a Cos a Formula in Terms of tung a 4 How to lớn Apply 2 Sin a Cos a Formula? 5 FAQs on 2 Sin a Cos a Formula

## What is 2 Sin a Cos a Formula?

2 sin a cos a formula is an important trigonometric formula that is equal lớn sin 2a. Mathematically, it is written as sin 2a = 2 sin a cos a. It can also be expressed in terms of rã a as well. The two ways in which 2 sin a cos a formula can be written are:

2 sin a cos a = sin 2a2 sin a cos a = (2 chảy a)/(1 + tan2a)

The first khung of this formula is the most commonly used form & it is used lớn simplify complex trigonometric functions và solving problems. We can also express 2 sin a cos a formula using sin2a + cos2a = 1 formula as:

2 sin a cos a = 2 √(1 - cos2a) cos a2 sin a cos a = 2 sin a √(1 - sin2a)

## Derivation of 2 Sin a Cos a Formula

Now, we will prove the 2 sin a cos a formula using the angle sum formula of the sine function, i.e., sin (a + b) = sin a cos b + sin b cos a. Assume a = b in this formula và let us derive the 2 sin a cos a formula step-wise. We have,

sin (a + b) = sin a cos b + sin b cos a

⇒ sin (a + a) = sin a cos a + sin a cos a

⇒ sin a cos a + sin a cos a = sin (a + a)

⇒ 2 sin a cos a = sin (2a)

Hence, we have proved that 2 sin a cos a is equal to lớn sin 2a.

## 2 Sin a Cos a Formula in Terms of tan a

Next, we will derive the formula of 2 sin a cos a in terms of tan a. We have derived that 2 sin a cos a = sin (2a). Now, if we multiply and divide 2 sin a cos a by cos a, then we have

2 sin a cos a = (2 sin a cos a) × (cos a)/(cos a)

= 2 (sin a/cos a) (cos2a)

Now, we know that sin x/cos x = tan x & 1/cos x = sec x or 1/sec x = cos x. Therefore, we have

2 sin a cos a = (2 chảy a)/(sec2a)

= (2 rã a)/(1 + tan2a)

Hence, the 2 sin a cos a formula in terms of tung a is given by, 2 sin a cos a = (2 tung a)/(1 + tan2a)

## How to Apply 2 Sin a Cos a Formula?

So far, we have derived the formula for 2 sin a cos a. Next, let us understand the application of these formulas in solving different problems. Let us solve a few examples to lớn learn how lớn apply 2 sin a cos a formula.

Example 1: Find the value of sin 120° using 2 sin a cos a formula.

Solution: We know the values of trigonometric functions for specific angles. So, we have

sin 120° = sin (2 × 60°)

⇒ sin 120° = 2 sin 60° cos 60° (Because 2 sin a cos a = sin (2a))

⇒ sin 120° = 2 × √3/2 × 1/2

⇒ sin 120° = √3/2

The formula can also be conversely used to find the value of 2 sin a cos a using sin 2a.

Xem thêm: Công Thức Gia Tốc Cực Đại - Xác Định Gia Tốc Cực Đại Trong Tổng Hợp Dao Động

Example 2: Determine the value of 2 sin 15° cos 15°.

Solution: As we know the values of sine function for specific angles & 2 sin a cos a = sin (2a), we have

2 sin 15° cos 15° = sin (2 × 15°)

⇒ 2 sin 15° cos 15° = sin 30°

⇒ 2 sin 15° cos 15° = 1/2

Important Notes on 2 sin a cos a

2 sin a cos a = sin (2a)2 sin a cos a = (2 rã a)/(1 + tan2a)2 sin a cos a = 2 √(1 - cos2a) cos a2 sin a cos a = 2 sin a √(1 - sin2a)

Related Topics on 2 sin a cos a